LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

Excerpts

LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The function $\phi(n)$ will be said to increase steadily with $n$ if $\phi(n + 1) \geq \phi(n)$ for all values of $n$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The commonest example of an infinite geometric series is given by an ordinary recurring decimal.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Thus every proper fraction can be expressed as a recurring decimal, and conversely.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
This number $M$ we call the *upper bound* of $S$, and we may enunciate the following theorem.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
the function which is equal to $1/(1 - x)$ if $-1 < x < 1$ and is undefined for all other values of $x$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader will find no difficulty in proving such theorems as the following, which are obvious extensions of theorems already proved for real functions and series.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
If $\phi(n)$ steadily increases, and $\psi(n)$ steadily decreases, as $n$ tends to $\infty$, and if $\psi(n) > \phi(n)$ for all values of $n$, then both $\phi(n)$ and $\psi(n)$ tend to limits, and $\lim\phi(n) \leq \lim\psi(n)$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
*Large* is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
On the other hand we cannot as a rule alter an *infinite* number of the values of $\phi(n)$ without affecting fundamentally its behaviour as $n$ tends to $\infty$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Were the problem however merely that of finding *some* function of $x$ to fulfil the condition stated, it would of course present no difficulty whatever.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
A function can only tend to $+\infty$ or to $-\infty$ if, after a certain value of $n$, it maintains a constant sign.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Take for instance the case of $k > 0$. Let $\Delta$ be any assigned number, however large.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
This number is evidently not divisible by any of $2$, $3$, $5$, … $N$, since the remainder when it is divided by any of these numbers is $1$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
There are however, as was first shown by Euclid, infinitely many primes.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Such an assertion is palpably absurd when made of a *fixed* number such as $\cos\frac{1}{2}\theta\pi$, which is not zero.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
When $\phi(n)$ does not tend to a limit, nor to $+\infty$, nor to $-\infty$, as $n$ tends to $\infty$, we say that $\phi(n)$ **** as $n$ tends to $\infty$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Oscillation is defined in a purely negative manner: a function oscillates when it does not do certain other things.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
We divide the real numbers $\xi$ into two classes $L$ and $R$, putting $\xi$ in $L$ or $R$ according as $\phi(n) \geq \xi$ for some value of $n$ (and so of course for all greater values), or $\phi(n) < \xi$ for all values of $n$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It is perhaps hardly necessary to point out that the theorem is not true if the condition that every $u_{n}$ is positive is not fulfilled.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It is a truism that in common life a number which is large in one connection is small in another; $6$ goals is a large score in a football match, but $6$ runs is not a large score in a cricket match; and $400$ runs is a large score, but £$400$ is not a large income: and so of course in mathematics *large* generally means *large enough*, and what is large enough for one purpose may not be large enough for another.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader cannot too strongly impress upon himself that when we say that $n$ ‘tends to $\infty$’ we mean simply that $n$ is supposed to assume a series of values which increase continually and without limit.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader should imagine himself confronted by an opponent who questions the truth of the statement. He would name a series of numbers growing smaller and smaller. He might begin with $.001$. The reader would reply that $1/n < .001$ as soon as $n > 1000$. The opponent would be bound to admit this, but would try again with some smaller number, such as $.000\MS000\MS1$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader cannot impress these facts too strongly on his mind. **limit is not a value of the function**: it is something quite distinct from these values, though it is defined by its relations to them and may possibly be equal to some of them.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The function $\phi(n)$ is said to tend to the limit $l$ as $n$ tends to $\infty$, if, however small be the positive number $\DELTA$, $\phi(n)$ differs from $l$ by less than $\DELTA$ for sufficiently large values of $n$; that is to say if, however small be the positive number $\DELTA$, we can determine a number $n_{0}(\DELTA)$ corresponding to $\DELTA$, such that $\phi(n)$ differs from $l$ by less than $\DELTA$ for all values of $n$ greater than or equal to $n_{0}(\DELTA)$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
If $n$ is any positive integer, such as $1000$, $1,000,000$ or any number we like to think of, then there are more than $n$ positive integers. Thus, if the number we think of is $1,000,000$, there are obviously at least $1,000,001$ positive integers.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
We may obviously alter the values of $\phi(n)$ for any finite number of values of $n$, in any way we please, without in the least affecting the behaviour of $\phi(n)$ as $n$ tends to $\infty$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
But now consider $\phi(n) = (-1)^{n}n$, the values of which are $-1$, $2$, $-3$, $4$, $-5$, …. This function oscillates, for it does not tend to a limit, nor to $+\infty$, nor to $-\infty$. And in this case we cannot assign any limit beyond which the numerical value of the terms does not rise.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It should be observed that in this case $\phi(2k + 1)$ is always less than $\phi(2k)$, so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however ‘oscillate’ according to our definition of the term.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Euclid’s proof is as follows. If there are only a finite number of primes, let them be $1$, $2$, $3$, $5$, $7$, $11$, … $N$. Consider the number $1 + (1 · 2 · 3 · 5 · 7 · 11 \dots N)$. This number is evidently not divisible by any of $2$, $3$, $5$, … $N$, since the remainder when it is divided by any of these numbers is $1$. It is therefore not divisible by any prime save $1$, and is therefore itself prime, which is contrary to our hypothesis.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
That a theorem is ‘obvious’ in this sense does not prove that it is true, since the most confident of the intuitive judgments of common sense are often found to be mistaken; and even if the theorem is true, the fact that it is also ‘obvious’ is no reason for not proving it, if a proof can be found. The object of mathematics is to prove that certain premises imply certain conclusions; and the fact that the conclusions may be as ‘obvious’ as the premises never detracts from the necessity, and often not even from the interest of the proof.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The argument which the reader will at once form in his mind is roughly this: ‘when $n$ is large, $\phi(n)$ is nearly equal to $a$ and $\psi(n)$ to $b$, and therefore their sum is nearly equal to $a + b$’. It is well to state the argument quite formally, however.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘$\phi(n)$ is very small’ for large values of $n$, we mean that we can choose $n_{0}$ so that $\phi(n)$ is numerically smaller than *any* assigned number, if $n$ is sufficiently large$n \geq n_{0}$. Such an assertion is palpably absurd when made of a *fixed* number such as $\cos\frac{1}{2}\theta\pi$, which is not zero.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
That is to say, while there are in general *five* alternatives as to the behaviour of a function, there are *two* only for this special kind of function.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of $n$ does or does not tend to a limit as $n \to \infty$, *without requiring us to be able to guess or otherwise infer beforehand what the limit is*.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It is to be observed that we do not exclude the case in which $\phi(n)$ has the *same* value for several values of $n$; all we exclude is possible *decrease*.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It should be noticed that the limit may be equal to $K$: if *e.g.* $\phi(n) = 3 - (1/n)$, then every value of $\phi(n)$ is less than $3$, but the limit is equal to $3$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Since $-\phi(n)$ always increases if $\phi(n)$ always decreases, it is not necessary to consider the two kinds of functions separately; for theorems proved for one kind can at once be extended to the other.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader may be tempted to think that the converse of the theorem is true and that if $\lim u_{n} = 0$ then the series $\sum u_{n}$ must be convergent. That this is not the case is easily seen from an example.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader should be warned that the words ‘divergent’ and ‘oscillatory’ are used differently by different writers.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Later on we shall be able to identify this function with the *Napierian logarithm* of $x$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The reader, if he desires to become expert in dealing with questions about limits, should study the argument above with great care. It is very often necessary, in proving the limit of some given expression to be zero, to split it into two parts which have to be proved to have the limit zero in slightly different ways. When this is the case the proof is never very easy.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The point of the proof is this: we have to prove that $(t_{1} + t_{2} + \dots + t_{n})/n$ is small when $n$ is large, the $t$’s being small when their suffixes are large. We split up the terms in the bracket into two groups. The terms in the first group are not all small, but their number is small compared with $n$. The number in the second group is *not* small compared with $n$, but the terms are all small, and their number at any rate less than $n$, so that their sum is small compared with $n$. Hence each of the parts into which $(t_{1} + t_{2} + \dots + t_{n})/n$ has been divided is small when $n$ is large.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
If $z^{n} \to l$ then $z^{n+1} \to l$, by (1) of [§]86. But, by (4) of [§]86, z^n+1 = zz^n zl, and therefore $l = zl$, which is only possible if (*a*) $l = 0$ or (*b*) $z = 1$. If $z = 1$ then $\lim z^{n} = 1$. Apart from this special case the limit, if it exists, can only be zero.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
If $\rho(n)$ and $\sigma(n)$ both converge to zero then it is plain that $\sqrtp{\rho^{2} + \sigma^{2}}$ does so. The converse follows from the fact that the numerical value of $\rho$ or $\sigma$ cannot be greater than $\sqrtp{\rho^{2} + \sigma^{2}}$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
If we change $\theta$ into $\theta + \pi$, we see that these results hold also for negative values of $r$ numerically less than $1$. Thus they hold when $-1 < r < 1$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
This example proves that the converse of Ex. 27 is not true: for $s_{n}$ oscillates as $n \to \infty$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
the series $1 + r + r^{2} + \dots$ diverges to $+\infty$ if $r \geq 1$, converges to $1/(1 - r)$ if $-1 < r < 1$, oscillates finitely if $r = -1$, and oscillates infinitely if $r < -1$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
For example, $.217 = .216\DPmod{\dot{9}}{\Repeat{9}}$. Thus every proper fraction can be expressed as a recurring decimal, and conversely.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
It also follows that every decimal which does not recur represents some *irrational* number between $0$ and $1$. Conversely, any such number can be expressed as such a decimal.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine $m$ and $p$ so that $m$, $m + p$, $m + 2p$, $m + 3p$, … are all prime numbers; and this is absurd, since the series includes $m + mp$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
An infinite aggregate of numbers does not necessarily possess a least member.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
This number $M$ is not exceeded by any member of $S$, but every number less than $M$ is exceeded by at least one member of $S$.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
The theoretical importance of the ‘general principle of convergence’ can hardly be overestimated. Like the theorems of [§]69, it gives us a means of deciding whether a function $\phi(n)$ tends to a limit or not, without requiring us to be able to tell beforehand what the limit, if it exists, must be; and it has not the limitations inevitable in theorems of such a special character as those of [§]69.

Equations

LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
y = \phi(n)
The function of the positive integer variable n is written as y = phi(n), with y regarded as a function of n defined for all values of n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
y = \phi(x) + \sin x\pi
A function of x that takes the value phi(n) at each positive integer x = n, because sin(n pi) = 0.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\sin n\pi = 0
The sine of n times pi vanishes for every integer n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
(-1)^{n} = \cos n\pi
For integer n, (-1) to the power n equals cos(n pi), giving a form of (-1)^n defined for all real x.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
y = 1 · 2 \dots n = n!
The product of the first n positive integers is written n!, which has no obvious formula in x that reduces to it at x = n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim_{n\to\infty} \frac{1}{n} = 0
The limit of 1/n as n tends to infinity is zero.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim_{n\to\infty} \left(1 - \frac{1}{n}\right) = 1
The limit of 1 - 1/n as n tends to infinity is one.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
1 - \phi(n) = 1/n
For phi(n) = 1 - 1/n, the difference 1 - phi(n) equals 1/n, which is why statement (ib) is true.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
n^2 \to \infty
n squared tends to infinity as n tends to infinity, meaning n squared is large for large n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
-n^{2} \to -\infty
Minus n squared tends to negative infinity as n tends to infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim_{n \to \infty} \phi(n) = l
phi(n) tends to the limit l as n tends to infinity: for any positive Delta, phi(n) differs from l by less than Delta for all n at or beyond some n_0(Delta).
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n) - l| < \DELTA
The distance of phi(n) from its limit l is less than Delta for all n greater than or equal to n_0(Delta); this is the test in Definition I.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) > \Delta
phi(n) exceeds any given number Delta for all n greater than or equal to n_0(Delta); this is the test in Definition II for tending to plus infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) \to +\infty
phi(n) tends to positive infinity as n tends to infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
n_{0} = 1 + [1/\DELTA]
For phi(n) = 1/n, the choice n_0 = 1 + [1/Delta] (with [x] the greatest integer not greater than x) satisfies the condition 1/n < Delta for n at or above n_0.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
1/n < \DELTA
For phi(n) = 1/n the inequality 1/n < Delta holds for all n greater than 1/Delta, so the sufficiently large values of n need only exceed 1/Delta.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
1000\{1 + (-1)^{n}\}/n < 1
The inequality 1000(1 + (-1)^n)/n < 1 holds for all n greater than 2000, the exceptions being the even values 2 to 2000.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = n^{k}
The function phi(n) is the power n^k, where k is a positive or negative integer or rational fraction.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim n^{k} = 0
If k is negative, n^k tends to the limit 0 as n tends to infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim n^{k} = 1
If k = 0, then n^k = 1 for every n, so the limit is 1.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = p_{n}
phi(n) is the n-th prime number p_n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) > n
For the prime-number function, phi(n) is greater than n for all n except n = 1, 2, 3.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = [\alpha n]
phi(n) is the integer part of alpha times n, where alpha is any positive number.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = 0\quad (0 \leq n < 1 / \alpha)
For alpha positive, the integer-part function phi(n) = [alpha n] is 0 while n lies between 0 and 1/alpha.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = 1/\{n - (-1)^{n}\}
One example function in the list of behaviours as n tends to infinity; it tends to 0.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n)| < 1/n
For phi(n) = sin(n theta pi)/n, the modulus of phi(n) is less than 1/n, because the sine is at most 1 in modulus.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = (-1)^{n}
The simplest oscillatory function: equal to +1 for even n and to -1 for odd n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = (-1)^{n} + (1/n)
An oscillating function whose values do not recur cyclically; every value is numerically at most 3/2.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = (-1)^{n}n
An oscillating function with no bound on its numerical value, so it oscillates infinitely.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim \sin n\theta\pi = l
Supposition tested in the argument: sin(n theta pi) tends to a limit l, which the argument shows is impossible for irrational theta.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\cos(n + \tfrac{1}{2})\theta\pi = \cos n\theta\pi \cos\tfrac{1}{2}\theta\pi - \sin n\theta\pi \sin\tfrac{1}{2}\theta\pi
The cosine of a sum of angles expands as cos a cos b minus sin a sin b, with a = n theta pi and b = theta pi / 2.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\cos(n - \tfrac{1}{2})\theta\pi = \cos n\theta\pi \cos\tfrac{1}{2}\theta\pi + \sin n\theta\pi \sin\tfrac{1}{2}\theta\pi
The cosine of a difference of angles expands as cos a cos b plus sin a sin b, with a = n theta pi and b = theta pi / 2.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\sin(np\pi/q) = (-1)^{ap}\sin(bp\pi/q)
With n = aq + b, the sine of n p pi / q equals (-1)^(ap) times the sine of b p pi / q.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n) + \psi(n) - a - b| < \DELTA
The sum phi(n) + psi(n) lies within any assigned positive number Delta of a + b once n is large enough.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\{\phi(n) + \psi(n)\} = a + b
If phi(n) and psi(n) tend to limits a and b, then phi(n) + psi(n) tends to a + b.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n) + \psi(n) - a - b| \leq |\phi(n) - a| + |\psi(n) - b|
The modulus of a sum is at most the sum of the moduli, which bounds the error of phi + psi by the errors of phi and psi.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n)\psi(n) = ab + a\psi_{1}(n) + b\phi_{1}(n) + \phi_{1}(n)\psi_{1}(n)
Writing phi = a + phi1 and psi = b + psi1, the product phi psi expands into ab plus three error terms.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\phi(n)\psi(n) = ab
If phi(n) tends to a and psi(n) tends to b, then the product phi(n) psi(n) tends to ab.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim k\phi(n) = ka
If phi(n) tends to a limit a, then k times phi(n) tends to k a, for a constant k.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\left|\frac{1}{\phi(n)} - \frac{1}{a}\right| = \frac{|\phi_{1}(n)|}{|a| |a + \phi_{1}(n)|}
The difference between 1/phi(n) and 1/a equals |phi1(n)| divided by |a| times |phi(n)|, which shows it becomes small when phi1 does.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\frac{1}{\phi(n)} = \frac{1}{a}
If phi(n) tends to a nonzero limit a, then 1/phi(n) tends to 1/a.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\frac{\phi(n)}{\psi(n)} = \frac{a}{b}
If phi(n) tends to a and psi(n) tends to a nonzero b, then phi(n)/psi(n) tends to a/b.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim R\{\phi(n), \psi(n), \chi(n), \dots\} = R(a, b, c, \dots)
If each of phi, psi, chi tends to a limit and the denominator of the rational function R does not vanish at those limits, then R tends to R evaluated at the limits.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
S(n) = \frac{a_{0}n^{p} + a_{1}n^{p-1} + \dots + a_{p}} {b_{0}n^{q} + b_{1}n^{q-1} + \dots + b_{q}}
The most general rational function of n, with leading coefficients a0 and b0 not zero, whose behaviour as n tends to infinity is studied.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim S(n) = 0\quad (p < q)
If the numerator's degree p is less than the denominator's degree q, S(n) tends to 0.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim S(n) = a_{0}/b_{0}\quad (p = q)
If the numerator and denominator have equal degree p = q, S(n) tends to the ratio a0/b0 of leading coefficients.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
u_{n} = r^{n-1}
The general term of the geometrical series is r raised to the power n-1.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} = 1 + r + r^{2} + \dots + r^{n-1} = (1 - r^{n})/(1 - r)
The sum of the first n terms of the geometrical series is (1 - r^n)/(1 - r), for r not equal to 1.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} = 1 + 1 + \dots + 1 = n
In the special case r = 1, the sum of the first n terms equals n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} \to +\infty
When r = 1 the partial sums s_n increase without limit.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} \geq n
If r is at least 1, the partial sum s_n is at least n, so s_n tends to plus infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
1/(1 - r)
The series 1 + r + r^2 + ... is convergent with this sum if and only if -1 < r < 1.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
f(x) = \lim_{n \to \infty} n(\sqrt[n]{x} - 1)
The function f(x) is defined as the limit as n tends to infinity of n times (the n-th root of x minus 1); this is the function encountered in Section 75.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
u_{1}(x) + u_{2}(x) + \dots = \lim_{n \to \infty}\{u_{1}(x) + u_{2}(x) + \dots + u_{n}(x)\}
The sum of an infinite series of functions of x is defined as the limit of its partial sums as n tends to infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim_{n \to \infty} \phi_{n}(x)
The limit as n tends to infinity of phi_n(x) is, for each x, a function of x; this is the function represented by the limit.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s \leq K
A set S of real numbers is bounded above if some number K satisfies s at most K for every member s of S.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s \geq k
A set S of real numbers is bounded below if some number k satisfies s at least k for every member s of S.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
k \leq m \leq M \leq K
When a set is bounded, its lower bound m and upper bound M lie between any lower bound k and any upper bound K.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) \leq M
The upper bound M of a bounded-above function phi(n) is not exceeded by any of its values.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) > M - \DELTA
For any positive number DELTA, some value of phi(n) exceeds M minus DELTA.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) \geq m
The lower bound m of a bounded-below function phi(n) is not exceeded from below by any of its values.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) < m + \DELTA
For any positive number DELTA, some value of phi(n) is less than m plus DELTA.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) \leq k
The book states the bounded-below condition with this inequality, but the lower bound is defined by values at least k; this appears to be a misprint for phi(n) >= k (flagged, not corrected).
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
m \leq \lambda \leq \Lambda \leq M
The lower and upper limits of indetermination lie between the lower and upper bounds of the function.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\Lambda = \limsup \phi(n)
The upper limit of indetermination of phi(n) as n tends to infinity is defined as the limit superior.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lambda = \liminf \phi(n)
The lower limit of indetermination of phi(n) as n tends to infinity is defined as the limit inferior.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) < \Lambda + \DELTA
For any positive DELTA, phi(n) is less than Lambda plus DELTA for all sufficiently large n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) > \Lambda - \DELTA
For any positive DELTA, phi(n) exceeds Lambda minus DELTA for infinitely many values of n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
l - \DELTA < \phi(n) < l + \DELTA
phi(n) tends to the limit l exactly when, for every positive DELTA, phi(n) lies between l minus DELTA and l plus DELTA for all sufficiently large n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\Lambda - \lambda \leq 2\DELTA
If phi(n) tends to l, the gap between the upper and lower limits of indetermination is at most 2 DELTA for every positive DELTA, so they are equal.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n_{2}) - \phi(n_{1})| < \DELTA
A bounded function tends to a limit if and only if, for any positive DELTA, phi(n_2) and phi(n_1) differ by less than DELTA whenever n_2 > n_1 >= n_0(DELTA).
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|u_{n_{1}+1} + u_{n_{1}+2} + \dots + u_{n_{2}}| < \DELTA
The series u_1 + u_2 + ... converges if and only if, for any positive DELTA, the sums of consecutive blocks of terms beyond some n_0 are all less than DELTA in absolute value.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n_{1}) - \DELTA < \phi(n_{2}) < \phi(n_{1}) + \DELTA
Fixing one value n_1 beyond n_0 bounds every later value of phi(n) within DELTA of phi(n_1), so phi(n) is bounded.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\rho(n) + i\sigma(n)
A complex function phi(n) is written as its real part rho(n) plus i times its imaginary part sigma(n), both real functions of n.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\phi(n) = l
A complex function phi(n) converges to l = r + is when its real and imaginary parts converge to r and s respectively.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
l = r + is
The complex limit l is the real part r plus i times the imaginary part s.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} = u_{1} + u_{2} + \dots + u_{n}
The partial sum s_n is the sum of the first n terms of the series.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
(v_{1} + v_{2} + \dots + v_{n}) + i(w_{1} + w_{2} + \dots + w_{n})
The partial sum of a complex series equals the sum of its real parts plus i times the sum of its imaginary parts, so it converges to l exactly when those real and imaginary series converge.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\phi(n + p) = l
If φ(n) tends to l, then φ(n + p) also tends to l for any fixed p.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim\{\phi(n) + \psi(n)\} = l + m
The limit of a sum of two sequences is the sum of their limits.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim k\phi(n) = kl
A constant k times a sequence tends to k times its limit.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim \phi(n)\psi(n) = lm
The limit of a product of two sequences is the product of their limits.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim u_{n} = 0
If the series u_1 + u_2 + u_3 + … converges, its terms u_n tend to zero.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|\phi(n)| = \sqrtbr{\{\rho(n)\}^{2} + \{\sigma(n)\}^{2}}
The modulus of a complex number φ(n) = ρ(n) + iσ(n) is the square root of the sum of the squares of its real and imaginary parts.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n)\psi(n) = \rho\rho' - \sigma\sigma' + i(\rho\sigma' + \rho'\sigma)
The product of two complex numbers is split into real and imaginary parts.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\phi(n) = \rho(n) + i\sigma(n)
A complex function φ(n) is written as its real part ρ(n) plus i times its imaginary part σ(n).
This equation is in COMPLEX NUMBERS (COMPLEX NUMBERS)
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
z^{n} = r^{n} (\cos n\theta + i\sin n\theta)
The n-th power of a complex number in polar form has modulus r^n and angle nθ.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
|z^{n}| = r^{n}
The modulus of z^n equals r^n, where r is the modulus of z.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
\lim z^{n} = 0
For complex z with modulus r < 1, z^n tends to zero as n tends to infinity.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
l = zl
If z^n tends to a limit l, then l equals zl, which forces l = 0 unless z = 1.
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
s_{n} = 1 + z + z^{2} + \dots + z^{n-1} = (1 - z^{n})/(1 - z)
The sum of the first n terms of the geometric series in z equals (1 - z^n)/(1 - z) when z is not 1.

Problems

Exercise XXIX

Exercise XXIX, problem 1, p. 143
**decimals.** The commonest example of an infinite geometric series is given by an ordinary recurring decimal. [pg]144 Consider, for example, the decimal $.217\DPmod{\dot{1}\dot{3}}{\Repeat{13}}$. This stands, according to the ordinary rules of arithmetic, for 210 + 110^2 + 710^3 + 110^4 + 310^5 + 110^6 + 310^7 + … = 2171000 + 1310^5 / (1 - 110^2) = 268712375. The reader should consider where and how any of the general theorems of [§]77 have been used in this reduction.
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Exercise XXIX, problem 2, p. 143
Show that in general .a_1a_2…a_m _1_2…_n _1_2…_n| = a_1a_2…a_m_1…_n - a_1a_2…a_n 99…900…0, the denominator containing $n$ $9$’s and $m$ $0$’s.
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Exercise XXIX, problem 3, p. 143
Show that a pure recurring decimal is always equal to a proper fraction whose denominator does not contain $2$ or $5$ as a factor.
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Exercise XXIX, problem 4, p. 143
A decimal with $m$ non-recurring and $n$ recurring decimal figures is equal to a proper fraction whose denominator is divisible by $2^{m}$ or $5^{m}$ but by no higher power of either.
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Exercise XXIX, problem 5, p. 143
The converses of Exs. 3, 4 are also true. Let $r = p/q$, and suppose first that $q$ is prime to $10$. If we divide all powers of $10$ by $q$ we can obtain at most $q$ different remainders. It is therefore possible to find two numbers $n_{1}$ and $n_{2}$, where $\DPtypo{n_{2} > n_{1}}{n_{1} > n_{2}}$, such that $10^{n_{1}}$ and $10^{n_{2}}$ give the same remainder. Hence $10^{n_{1}} - 10^{n_{2}} = 10^{n_{2}}(10^{n_{1}-n_{2}} - 1)$ is divisible by $q$, and so $10^{n} - 1$, where $n = n_{1} - n_{2}$, is divisible by $q$. Hence $r$ may be expressed in the form $P/(10^{n} - 1)$, or in the form P10^n + P10^2n + …, *i.e.* as a pure recurring decimal with $n$ figures. If on the other hand $q = 2^{\alpha}5^{\beta}Q$, where $Q$ is prime to $10$, and $m$ is the greater of $\alpha$ and $\beta$, then $10^{m}r$ has a denominator prime to $10$, and is therefore expressible as the sum of an integer and a pure recurring decimal. But this is not true of $10^{\mu}r$, for any value of $\mu$ less than $m$; hence the decimal for $r$ has exactly $m$ non-recurring figures.
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Exercise XXIX, problem 6, p. 143
To the results of Exs. 2--5 we must add that of % [examples:i]Ex. i%. 3. Finally, if we observe that .99| = 910 + 910^2 + 910^3 + … = 1, we see that every terminating decimal can also be expressed as a mixed recurring decimal whose recurring part is composed entirely of $9$’s. For example, $.217 = .216\DPmod{\dot{9}}{\Repeat{9}}$. Thus every proper fraction can be expressed as a recurring decimal, and conversely.
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Exercise XXIX, problem 7, p. 143
**in general. The expression of irrational numbers as non-recurring decimals.** Any decimal, whether recurring or not, corresponds to a definite number between $0$ and $1$. For the decimal $.a_{1}a_{2}a_{3}a_{4}\dots$ stands for the series a_110 + a_210^2 + a_310^3 + …. [pg]145 Since all the digits $a_{r}$ are positive, the sum $s_{n}$ of the first $n$ terms of this series increases with $n$, and it is certainly not greater than $.\DPmod{\dot{9}}{\Repeat{9}}$ or $1$. Hence $s_{n}$ tends to a limit between $0$ and $1$. Moreover no two decimals can correspond to the same number (except in the special case noticed in Ex. 6). For suppose that $.a_{1}a_{2}a_{3} \dots$, $.b_{1}b_{2}b_{3} \dots$ are two decimals which agree as far as the figures $a_{r-1}$, $b_{r-1}$, while $a_{r} > b_{r}$. Then $a_{r}\geq b_{r} + 1 > b_{r}.b_{r+1}b_{r+2} \dots$ (unless $b_{r+1}$, $b_{r+2}$, … are all $9$’s), and so .a_1a_2 …a_ra_r+1 …> .b_1b_2 …b_rb_r+1 …. It follows that the expression of a rational fraction as a recurring decimal (Exs. 2--6) is unique. It also follows that every decimal which does not recur represents some *irrational* number between $0$ and $1$. Conversely, any such number can be expressed as such a decimal. For it must lie in one of the intervals 0, 1/10;0pt minus 3pt1/10, 2/10; …;0pt minus 3pt9/10, 1. If it lies between $r/10$ and $(r + 1)/10$, then the first figure is $r$. By subdividing this interval into $10$ parts we can determine the second figure; and so on. But (Exs. 3, 4) the decimal cannot recur. Thus, for example, the decimal $1.414\dots$, obtained by the ordinary process for the extraction of $\sqrt{2}$, cannot recur.
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Exercise XXIX, problem 8, p. 143
The decimals $.101\MS001\MS000\MS100\MS001\MS0\dots$ and $.202\MS002\MS000\MS200\MS002\MS0\dots$, in which the number of zeros between two $1$’s or $2$’s increases by one at each stage, represent irrational numbers.
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Exercise XXIX, problem 9, p. 143
The decimal $.111\MS010\MS100\MS010\MS10\dots$, in which the $n$th figure is $1$ if $n$ is prime, and zero otherwise, represents an irrational number. [Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine $m$ and $p$ so that $m$, $m + p$, $m + 2p$, $m + 3p$, … are all prime numbers; and this is absurd, since the series includes $m + mp$.] All the results of xxix may be extended, with suitable modifications, to decimals in any scale of notation. For a fuller discussion see Bromwich, *Infinite Series*, Appendix I.
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Exercise XXX

Exercise XXX, problem 1, p. 145
0.375em plus 0.75em minus 0.25emThe series $r^{m} + r^{m+1} + \dots$ is convergent if $-1 < r < 1$, and its sum is $1/(1 - r) - 1 - r - \dots - r^{m-1}$ ([§]77, (2)).
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Exercise XXX, problem 10a, p. 145
Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.
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Exercise XXX, problem 10b, p. 145
Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.
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Exercise XXX, problem 10c, p. 145
Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.
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Exercise XXX, problem 10d, p. 145
Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.
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Exercise XXX, problem 11, p. 145
If $0 \leq a_{n} \leq 1$ then the series $a_{0} + a_{1}r + a_{2}r^{2} + \dots$ is convergent for $0 \leq r < 1$, and its sum is not greater than $1/(1 - r)$.
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Exercise XXX, problem 12, p. 145
If in addition the series $a_{0} + a_{1} + a_{2} + \dots$ is convergent, then the series $a_{0} + a_{1}r + a_{2}r^{2} + \dots$ is convergent for $0 \leq r \leq 1$, and its sum is not greater than the lesser of $a_{0} + a_{1} + a_{2} + \dots$ and $1/(1 - r)$.
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Exercise XXX, problem 13, p. 145
The series 1 + 11 + 11·2 + 11·2·3 + … is convergent. [For $1/(1·2 \dots n) \leq 1/2^{n-1}$.]
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Exercise XXX, problem 14a, p. 145
The series 1 + 11·2 + 11·2·3·4 + …,0pt minus 3pt11 + 11·2·3 + 11·2·3·4·5 + … are convergent.
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Exercise XXX, problem 14b, p. 145
The series 1 + 11·2 + 11·2·3·4 + …,0pt minus 3pt11 + 11·2·3 + 11·2·3·4·5 + … are convergent.
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Exercise XXX, problem 15, p. 145
The general harmonic series 1a + 1a + b + 1a + 2b + …, where $a$ and $b$ are positive, diverges to $+\infty$. [For $u_{n} = 1/(a + nb) > 1/\{n(a + b)\}$. Now compare with $1 + \frac{1}{2} + \frac{1}{3} + \dots$.]
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Exercise XXX, problem 16, p. 145
Show that the series (u_0 - u_1) + (u_1 - u_2) + (u_2 - u_3) + … is convergent if and only if $u_{n}$ tends to a limit as $n \to \infty$.
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Exercise XXX, problem 17, p. 145
If $u_{1} + u_{2} + u_{3} + \dots$ is divergent then so is any series formed by grouping the terms in brackets in any way to form new single terms.
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Exercise XXX, problem 18, p. 145
Any series, formed by taking a selection of the terms of a convergent series of positive terms, is itself convergent.
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Exercise XXX, problem 2, p. 145
The series $r^{m} + r^{m+1} + \dots$ is convergent if $-1 < r < 1$, and its sum is $r^{m}/(1 - r)$ ([§]77, (4)). Verify that the results of Exs. 1 and 2 are in agreement.
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Exercise XXX, problem 3a, p. 145
Prove that the series $1 + 2r + 2r^{2} + \dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \dots)$, ($\beta$) by writing it in the form $1 + 2(r + r^{2} + \dots)$, ($\gamma$) by adding the two series $1 + r + r^{2} + \dots$, $r + r^{2} + \dots$. In each case mention which of the theorems of [§]77 are used in your proof.
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Exercise XXX, problem 3b, p. 145
Prove that the series $1 + 2r + 2r^{2} + \dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \dots)$, ($\beta$) by writing it in the form $1 + 2(r + r^{2} + \dots)$, ($\gamma$) by adding the two series $1 + r + r^{2} + \dots$, $r + r^{2} + \dots$. In each case mention which of the theorems of [§]77 are used in your proof.
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Exercise XXX, problem 3c, p. 145
Prove that the series $1 + 2r + 2r^{2} + \dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \dots)$, ($\beta$) by writing it in the form $1 + 2(r + r^{2} + \dots)$, ($\gamma$) by adding the two series $1 + r + r^{2} + \dots$, $r + r^{2} + \dots$. In each case mention which of the theorems of [§]77 are used in your proof.
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Exercise XXX, problem 4, p. 145
Prove that the ‘arithmetic’ series a + (a + b) + (a + 2b) + … is always divergent, unless both $a$ and $b$ are zero. Show that, if $b$ is not zero, the series diverges to $+\infty$ or to $-\infty$ according to the sign of $b$, while if $b = 0$ it diverges to $+\infty$ or $-\infty$ according to the sign of $a$.
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Exercise XXX, problem 5, p. 145
What is the sum of the series (1 - r) + (r - r^2) + (r^2 - r^3) + … when the series is convergent? [The series converges only if $-1 < r \leq 1$. Its sum is $1$, except when $r = 1$, when its sum is $0$.]
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Exercise XXX, problem 6, p. 145
Sum the series %[** TN: In-line equation in the original] r^2 + r^21 + r^2 + r^2(1 + r^2)^2 + …. [The series is always convergent. Its sum is $1 + r^{2}$, except when $r = 0$, when its sum is $0$.]
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Exercise XXX, problem 7, p. 145
If we assume that $1 + r + r^{2} + \dots$ is convergent then we can prove that its sum is $1/(1 - r)$ by means of [§]77, (1) and (4). For if $1 + r + r^{2} + \dots = s$ then s = 1 + r(1 + r^2 + …) = 1 + rs.
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Exercise XXX, problem 8, p. 145
Sum the series r + r1 + r + r(1 + r)^2 + … when it is convergent. [The series is convergent if $-1 < 1/(1 + r) < 1$, *i.e.* if $r < -2$ or if $r > 0$, and its sum is $1 + r$. It is also convergent when $r = 0$, when its sum is $0$.]
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Exercise XXX, problem 9a, p. 145
Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*
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Exercise XXX, problem 9b, p. 145
Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXX, problem 9c, p. 145
Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXX, problem 9d, p. 145
Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise XXXI

Exercise XXXI, problem 1, p. 148
$\phi_{n}(x) = x$. Here $n$ does not appear at all in the expression of $\phi_{n}(x)$, and $\phi(x) = \lim\phi_{n}(x) = x$ for all values of $x$.
Printed answer:
- $\phi_{n}(x) = x$. Here $n$ does not appear at all in the expression of $\phi_{n}(x)$, and $\phi(x) = \lim\phi_{n}(x) = x$ for all values of $x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 10a, p. 148
$\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
Printed answer:
- $\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 10b, p. 148
$\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
Printed answer:
- $\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 10c, p. 148
$\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
Printed answer:
- $\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\phi(x) = 1$ when $|x| > 1$, $\phi(x) = -1$ when $|x| < 1$, $\phi(x) = 0$ when $x = 1$ and $\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 11, p. 148
Construct an example in which $\phi(x) = 1$, ($|x| > 1$); $\phi(x) = -1$, ($|x| < 1$); and $\phi(x) = 0$, ($x = 1$ and $x = -1$).
Printed answer:
- Construct an example in which $\phi(x) = 1$, ($|x| > 1$); $\phi(x) = -1$, ($|x| < 1$); and $\phi(x) = 0$, ($x = 1$ and $x = -1$).
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 12a, p. 148
$\phi_{n}(x) = x\{(x^{2n} - 1)/(x^{2n} + 1)\}^{2}$, $n/(x^{n} + x^{-n} + n)$.
Printed answer:
- $\phi_{n}(x) = x\{(x^{2n} - 1)/(x^{2n} + 1)\}^{2}$, $n/(x^{n} + x^{-n} + n)$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 12b, p. 148
$\phi_{n}(x) = x\{(x^{2n} - 1)/(x^{2n} + 1)\}^{2}$, $n/(x^{n} + x^{-n} + n)$.
Printed answer:
- $\phi_{n}(x) = x\{(x^{2n} - 1)/(x^{2n} + 1)\}^{2}$, $n/(x^{n} + x^{-n} + n)$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 13, p. 148
$\phi_{n}(x) = \{x^{n}f(x) + g(x)\}/(x^{n} + 1)$. [Here $\phi(x) = f(x)$, ($|x| > 1$); $\phi(x) = g(x)$, ($|x| < 1$); $\phi(x) = \frac{1}{2}\{f(x) + g(x)\}$, ($x = 1$); and $\phi(x)$ is undefined when $x = -1$.]
Printed answer:
- $\phi_{n}(x) = \{x^{n}f(x) + g(x)\}/(x^{n} + 1)$. [Here $\phi(x) = f(x)$, ($|x| > 1$); $\phi(x) = g(x)$, ($|x| < 1$); $\phi(x) = \frac{1}{2}\{f(x) + g(x)\}$, ($x = 1$); and $\phi(x)$ is undefined when $x = -1$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 14, p. 148
$\phi_{n}(x) = (2/\pi) \arctan(nx)$. [$\phi(x) = 1$, ($x > 0$); $\phi(x) = 0$, ($x = 0$); $\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers, and is usually denoted by $\sgn x$.]
Printed answer:
- $\phi_{n}(x) = (2/\pi) \arctan(nx)$. [$\phi(x) = 1$, ($x > 0$); $\phi(x) = 0$, ($x = 0$); $\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers, and is usually denoted by $\sgn x$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 15, p. 148
$\phi_{n}(x) = \sin nx\pi$. [$\phi(x) = 0$ when $x$ is an integer; and $\phi(x)$ is otherwise undefined (% [examples:xxiv]Ex. xxiv%. 7).]
Printed answer:
- $\phi_{n}(x) = \sin nx\pi$. [$\phi(x) = 0$ when $x$ is an integer; and $\phi(x)$ is otherwise undefined (% [examples:xxiv]Ex. xxiv%. 7).]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 16, p. 148
If $\phi_{n}(x) = \sin (n!\, x\pi)$ then $\phi(x) = 0$ for all rational values of $x$ (% [examples:xxiv]Ex. xxiv%. 14). [The consideration of irrational values presents greater difficulties.]
Printed answer:
- If $\phi_{n}(x) = \sin (n!\, x\pi)$ then $\phi(x) = 0$ for all rational values of $x$ (% [examples:xxiv]Ex. xxiv%. 14). [The consideration of irrational values presents greater difficulties.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 17, p. 148
$\phi_{n}(x) = (\cos^{2} x\pi)^{n}$. [$\phi(x) = 0$ except when $x$ is integral, when $\phi(x) = 1$.]
Printed answer:
- $\phi_{n}(x) = (\cos^{2} x\pi)^{n}$. [$\phi(x) = 0$ except when $x$ is integral, when $\phi(x) = 1$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 18, p. 148
If $N \geq 1752$ then the number of days in the year $N$ a.d. is 365 + (^2 14 N)^n - (^2 1100 N)^n + (^2 1400 N)^n.
Printed answer:
- If $N \geq 1752$ then the number of days in the year $N$ a.d. is 365 + (^2 14 N)^n - (^2 1100 N)^n + (^2 1400 N)^n.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 2, p. 148
$\phi_{n}(x) = x/n$. Here $\phi(x) = \lim\phi_{n}(x) = 0$ for all values of $x$.
Printed answer:
- $\phi_{n}(x) = x/n$. Here $\phi(x) = \lim\phi_{n}(x) = 0$ for all values of $x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 3, p. 148
$\phi_{n}(x) = nx$. If $x > 0$, $\phi_{n}(x) \to +\infty$; if $x < 0$, $\phi_{n}(x) \to -\infty$: only when $x = 0$ has $\phi_{n}(x)$ a limit (viz. $0$) as $n \to \infty$. Thus $\phi(x) = 0$ when $x = 0$ and is not defined for any other value of $x$.
Printed answer:
- $\phi_{n}(x) = nx$. If $x > 0$, $\phi_{n}(x) \to +\infty$; if $x < 0$, $\phi_{n}(x) \to -\infty$: only when $x = 0$ has $\phi_{n}(x)$ a limit (viz. $0$) as $n \to \infty$. Thus $\phi(x) = 0$ when $x = 0$ and is not defined for any other value of $x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 4a, p. 148
$\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.
Printed answer:
- $\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 4b, p. 148
$\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.
Printed answer:
- $\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 5, p. 148
$\phi_{n}(x) = x^{n}$. Here $\phi(x) = 0$, ($-1 < x < 1$); $\phi(x) = 1$, ($x = 1$); and $\phi(x)$ is not defined for any other value of $x$.
Printed answer:
- $\phi_{n}(x) = x^{n}$. Here $\phi(x) = 0$, ($-1 < x < 1$); $\phi(x) = 1$, ($x = 1$); and $\phi(x)$ is not defined for any other value of $x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 6, p. 148
$\phi_{n}(x) = x^{n}(1 - x)$. Here $\phi(x)$ differs from the $\phi(x)$ of Ex. 5 in that it has the value $0$ when $x = 1$.
Printed answer:
- $\phi_{n}(x) = x^{n}(1 - x)$. Here $\phi(x)$ differs from the $\phi(x)$ of Ex. 5 in that it has the value $0$ when $x = 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 7, p. 148
$\phi_{n}(x) = x^{n}/n$. Here $\phi(x)$ differs from the $\phi(x)$ of Ex. 6 in that it has the value $0$ when $x = -1$ as well as when $x = 1$.
Printed answer:
- $\phi_{n}(x) = x^{n}/n$. Here $\phi(x)$ differs from the $\phi(x)$ of Ex. 6 in that it has the value $0$ when $x = -1$ as well as when $x = 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 8, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\phi(x) = 0$, ($-1 < x < 1$); $\phi(x) = \frac{1}{2}$, ($x = 1$); $\phi(x) = 1$, ($x < -1$ or $x > 1$); and $\phi(x)$ is not defined when $x = -1$.]
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\phi(x) = 0$, ($-1 < x < 1$); $\phi(x) = \frac{1}{2}$, ($x = 1$); $\phi(x) = 1$, ($x < -1$ or $x > 1$); and $\phi(x)$ is not defined when $x = -1$.]
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 9a, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 9b, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 9c, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 9d, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXI, problem 9e, p. 148
$\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
Printed answer:
- $\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise XXXII

Exercise XXXII, problem 1, p. 151
Neither $\Lambda$ nor $\lambda$ is affected by any alteration in any finite number of values of $\phi(n)$.
Printed answer:
- (none printed)
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How it was checked
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Exercise XXXII, problem 2, p. 151
If $\phi(n) = a$ for all values of $n$, then $m = \lambda = \Lambda = M = a$.
Printed answer:
- (none printed)
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How it was checked
- other: not a kind the checker handles
Exercise XXXII, problem 3, p. 151
If $\phi(n) = 1/n$, then $m = \lambda = \Lambda = 0$ and $M = 1$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXII, problem 4, p. 151
If $\phi(n) = (-1)^{n}$, then $m = \lambda = -1$ and $\Lambda = M = 1$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXII, problem 5, p. 151
If $\phi(n) = (-1)^{n}/n$, then $m = -1$, $\lambda = \Lambda = 0$, $M = \frac{1}{2}$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise XXXII, problem 6, p. 151
If $\phi(n) = (-1)^{n}\{1 + (1/n)\}$, then $m = -2$, $\lambda = -1$, $\Lambda = 1$, $M = \frac{3}{2}$.
Printed answer:
- (none printed)
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How it was checked
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Exercise XXXII, problem 7, p. 151
Let $\phi(n) = \sin n\theta\pi$, where $\theta > 0$. If $\theta$ is an integer then $m = \lambda = \Lambda = M = 0$. If $\theta$ is rational but not integral a variety of cases arise. Suppose, *e.g.*, that $\theta = p/q$, $p$ and $q$ being positive, odd, and prime to one another, and $q > 1$. Then $\phi(n)$ assumes the cyclical sequence of values (p/q),0pt minus 3pt(2p/q), …,0pt minus 3pt(2q - 1)p/q,0pt minus 3pt(2qp/q), …. It is easily verified that the numerically greatest and least values of $\phi(n)$ are $\cos(\pi/2q)$ and $-\cos(\pi/2q)$, so that m = = -(/2q),0pt minus 3pt= M = (/2q). The reader may discuss similarly the cases which arise when $p$ and $q$ are not both odd. The case in which $\theta$ is irrational is more difficult: it may be shown that in this case $m = \lambda = -1$ and $\Lambda = M = 1$. It may also be shown that the values of $\phi(n)$ are scattered all over the interval $\DPmod{(-1, 1)}{[-1, 1]}$ in such a way that, if $\xi$ is [pg]152 *any* number of the interval, then there is a sequence $n_{1}$, $n_{2}$, … such that $\phi(n_{k}) \to \xi$ as $k \to \infty$. A number of simple proofs of this result are given by Hardy and Littlewood, “Some Problems of Diophantine Approximation”, *Acta Mathematica*, vol. xxxvii. The results are very similar when $\phi(n)$ is the fractional part of $n\theta$.
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How it was checked
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Exercise XXIII

Exercise XXIII, problem 1, p. 120
$\phi(n) = n^{k}$, where $k$ is a positive or negative integer or rational fraction.
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Exercise XXIII, problem 2, p. 120
$\phi(n) = p_{n}$, where $p_{n}$ is the $n$th prime number.
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How it was checked
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Exercise XXIII, problem 3, p. 120
Let $\phi(n)$ be the number of primes less than $n$.
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Exercise XXIII, problem 4, p. 120
$\phi(n) = [\alpha n]$, where $\alpha$ is any positive number.
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How it was checked
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Exercise XXIII, problem 5, p. 120
If $\phi(n) = 1\MC000\MC000/n$, then $\lim\phi(n) = 0$: and if $\psi(n) = n/1\MC000\MC000$, then $\psi(n) \to +\infty$.
Printed answer:
- (none printed)
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How it was checked
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Exercise XXIII, problem 6, p. 120
$\phi(n) = 1/\{n - (-1)^{n}\}$, $n - (-1)^{n}$, $n\{1 - (-1)^{n}\}$.
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How it was checked
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Exercise XXIII, problem 7, p. 120
$\phi(n) = (\sin n\theta\pi)/n$, where $\theta$ is any real number.
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How it was checked
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Exercise XXIII, problem 8, p. 120
$\phi(n) = (\sin n\theta\pi)/\sqrt{n}$, $(a\cos^{2} n\theta + b\sin^{2}n\theta)/n$, where $a$ and $b$ are any real numbers.
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Exercise XXIII, problem 9, p. 120
$\phi(n) = \sin n\theta\pi$.
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Exercise XXIV

Exercise XXIV, problem 10, p. 122
$a + bn + (-1)^{n} (c + dn) + e\cos n\theta\pi + f\sin n\theta\pi$.
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How it was checked
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Exercise XXIV, problem 11, p. 122
$n\sin n\theta\pi$. If $\DPtypo{n}{\theta}$ is integral, then $\phi(n) = 0$, $\phi(n) \to 0$. If $\theta$ is rational but not integral, or irrational, then $\phi(n)$ oscillates infinitely.
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How it was checked
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Exercise XXIV, problem 12, p. 122
$n(a\cos^{2} n\theta\pi + b\sin^{2} n\theta\pi)$. In this case $\phi(n)$ tends to $+\infty$ if $a$ and $b$ are both positive, but to $-\infty$ if both are negative. Consider the special cases in which $a = 0$, $b > 0$, or $a > 0$, $b = 0$, or $a = 0$, $b = 0$. If $a$ and $b$ have opposite signs $\phi(n)$ generally oscillates infinitely. Consider any exceptional cases.
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Exercise XXIV, problem 13, p. 122
$\sin(n^{2}\theta\pi)$. If $\theta$ is integral, then $\phi(n) \to 0$. Otherwise $\phi(n)$ oscillates finitely, as may be shown by arguments similar to though more complex than those used in xxiii. 9 and []xxiv. 7. See Bromwich’s *Infinite Series*, p. 485.
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Exercise XXIV, problem 14, p. 122
$\sin(n!\, \theta\pi)$. If $\theta$ has a rational value $p/q$, then $n!\, \theta$ is certainly integral for all values of $n$ greater than or equal to $q$. Hence $\phi(n) \to 0$. The case in which $\theta$ is irrational cannot be dealt with without the aid of considerations of a much more difficult character.
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Exercise XXIV, problem 15a, p. 122
$\cos(n!\, \theta\pi)$, $a\cos^{2}(n!\, \theta\pi) + b\sin^{2}(n!\, \theta\pi)$, where $\theta$ is rational.
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Exercise XXIV, problem 15b, p. 122
$\cos(n!\, \theta\pi)$, $a\cos^{2}(n!\, \theta\pi) + b\sin^{2}(n!\, \theta\pi)$, where $\theta$ is rational.
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Exercise XXIV, problem 16a, p. 122
$an - [bn]$, $(-1)^{n}(an - [bn])$.
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Exercise XXIV, problem 16b, p. 122
$an - [bn]$, $(-1)^{n}(an - [bn])$.
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Exercise XXIV, problem 17a, p. 122
$[\sqrt{n}]$, $(-1)^{n}[\sqrt{n}]$, $\sqrt{n} - [\sqrt{n}]$.
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Exercise XXIV, problem 17b, p. 122
$[\sqrt{n}]$, $(-1)^{n}[\sqrt{n}]$, $\sqrt{n} - [\sqrt{n}]$.
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Exercise XXIV, problem 17c, p. 122
$[\sqrt{n}]$, $(-1)^{n}[\sqrt{n}]$, $\sqrt{n} - [\sqrt{n}]$.
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Exercise XXIV, problem 18, p. 122
*The smallest prime factor of $n$*. When $n$ is a prime, $\phi(n) = n$. When $n$ is even, $\phi(n) = 2$. Thus $\phi(n)$ oscillates infinitely.
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Exercise XXIV, problem 19, p. 122
*The largest prime factor of $n$*.
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Exercise XXIV, problem 1a, p. 122
$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\MC000\MC000/n) + (-1)^{n}$, $1\MC000\MC000(-1)^{n} + (1/n)$.
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Exercise XXIV, problem 1b, p. 122
$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\MC000\MC000/n) + (-1)^{n}$, $1\MC000\MC000(-1)^{n} + (1/n)$.
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Exercise XXIV, problem 1c, p. 122
$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\MC000\MC000/n) + (-1)^{n}$, $1\MC000\MC000(-1)^{n} + (1/n)$.
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Exercise XXIV, problem 1d, p. 122
$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\MC000\MC000/n) + (-1)^{n}$, $1\MC000\MC000(-1)^{n} + (1/n)$.
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Exercise XXIV, problem 20, p. 122
*The number of days in the year $n$ a.d.*
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Exercise XXIV, problem 2a, p. 122
$(-1)^{n}n$, $1\MC000\MC000 + (-1)^{n}n$.
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Exercise XXIV, problem 2b, p. 122
$(-1)^{n}n$, $1\MC000\MC000 + (-1)^{n}n$.
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Exercise XXIV, problem 3a, p. 122
$1\MC000\MC000 - n$, $(-1)^{n}(1\MC000\MC000 - n)$.
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Exercise XXIV, problem 3b, p. 122
$1\MC000\MC000 - n$, $(-1)^{n}(1\MC000\MC000 - n)$.
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Exercise XXIV, problem 4, p. 122
$n\{1 + (-1)^{n}\}$. In this case the values of $\phi(n)$ are 0,0pt minus 3pt4,0pt minus 3pt0,0pt minus 3pt8,0pt minus 3pt0,0pt minus 3pt12,0pt minus 3pt0,0pt minus 3pt16, …. The odd terms are all zero and the even terms tend to $+\infty$: $\phi(n)$ oscillates infinitely.
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Exercise XXIV, problem 5, p. 122
$n^{2} + (-1)^{n}2n$. The second term oscillates infinitely, but the first is very much larger than the second when $n$ is large. In fact $\phi(n) \geq n^{2} - 2n$ and $n^{2} - 2n = (n - 1)^{2} - 1$ is greater than any assigned value $\Delta$ if $n > 1 + \sqrtp{\Delta + 1}$. Thus $\phi(n) \to +\infty$. It should be observed that in this case $\phi(2k + 1)$ is always less than $\phi(2k)$, so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however ‘oscillate’ according to our definition of the term.
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Exercise XXIV, problem 6a, p. 122
$n^{2}\{1 + (-1)^{n}\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.
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Exercise XXIV, problem 6b, p. 122
$n^{2}\{1 + (-1)^{n}\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.
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Exercise XXIV, problem 6c, p. 122
$n^{2}\{1 + (-1)^{n}\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.
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Exercise XXIV, problem 7, p. 122
$\sin n\theta\pi$. We have already seen (xxiii. 9) that $\phi(n)$ oscillates finitely when $\theta$ is rational, unless $\theta$ is an integer, when $\phi(n)= 0$, $\phi(n) \to 0$. The case in which $\theta$ is irrational is a little more difficult. But it is not difficult to see that $\phi(n)$ still oscillates finitely. We can without loss of generality suppose $0 < \theta < 1$. In the first place $|\phi(n)| < 1$. Hence $\phi(n)$ must oscillate finitely or tend to a limit. We shall consider whether the second alternative is really possible. Let us suppose that n= l. 0.375em plus 0.75em minus 0.25emThen, however small $\DELTA$ may be, we can choose $n_{0}$ so that $\sin n\theta\pi$ lies between $l - \DELTA$ and $l + \DELTA$ for all values of $n$ greater than or equal to $n_{0}$. Hence $\sin(n + 1)\theta\pi - \sin n\theta\pi$ is numerically less than $2\DELTA$ for all such values of $n$, and so $|\sin \frac{1}{2}\theta\pi \cos(n + \frac{1}{2})\theta\pi| < \DELTA$. Hence (n + 12) = n12 - n12 must be numerically less than $\DELTA/|\sin\frac{1}{2}\theta\pi|$. Similarly (n - 12) = n12 + n12 must be numerically less than $\DELTA/|\sin\frac{1}{2}\theta\pi|$; and so each of $\cos n\theta\pi \cos\frac{1}{2}\theta\pi$, $\sin n\theta\pi \sin\frac{1}{2}\theta\pi$ must be numerically less than $\DELTA/|\sin\frac{1}{2}\theta\pi|$. That is to say, $\cos n\theta\pi \cos\frac{1}{2}\theta\pi$ is very small if $n$ is large, and this can only be the case if $\cos n\theta\pi$ is very small. Similarly $\sin n\theta\pi$ must be very small, so that $l$ must be zero. But it is impossible that $\cos n\theta\pi$ and $\sin n\theta\pi$ can *both* be very small, as the sum of their squares is unity. Thus the hypothesis that $\sin n\theta\pi$ tends to a limit $l$ is impossible, and therefore $\sin n\theta\pi$ oscillates as $n$ tends to $\infty$. 0.375em plus 0.75em minus 0.25emThe reader should consider with particular care the argument ‘$\cos n\theta\pi \cos\frac{1}{2}\theta\pi$ is very small, and this can only be the case if $\cos n\theta\pi$ is very small’. Why, he may ask, should it not be the other factor $\cos\frac{1}{2}\theta\pi$ which is ‘very small’? The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘$\phi(n)$ is very small’ for large values of $n$, we mean that we can choose $n_{0}$ so that $\phi(n)$ is numerically smaller than *any* assigned number, if $n$ is sufficiently large$n \geq n_{0}$. Such an assertion is palpably absurd when made of a *fixed* number such as $\cos\frac{1}{2}\theta\pi$, which is not zero. Prove similarly that $\cos n\theta\pi$ oscillates finitely, unless $\theta$ is an even integer.
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Exercise XXIV, problem 8a, p. 122
$\sin n\theta\pi + (1/n)$, $\sin n\theta\pi + 1$, $\sin n\theta\pi + n$, $(-1)^{n} \sin n\theta\pi$.
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Exercise XXIV, problem 8b, p. 122
$\sin n\theta\pi + (1/n)$, $\sin n\theta\pi + 1$, $\sin n\theta\pi + n$, $(-1)^{n} \sin n\theta\pi$.
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Exercise XXIV, problem 8c, p. 122
$\sin n\theta\pi + (1/n)$, $\sin n\theta\pi + 1$, $\sin n\theta\pi + n$, $(-1)^{n} \sin n\theta\pi$.
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Exercise XXIV, problem 8d, p. 122
$\sin n\theta\pi + (1/n)$, $\sin n\theta\pi + 1$, $\sin n\theta\pi + n$, $(-1)^{n} \sin n\theta\pi$.
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Exercise XXIV, problem 9a, p. 122
$a\cos n\theta\pi + b\sin n\theta\pi$, $\sin^{2}n\theta\pi$, $a\cos^{2}n\theta\pi + b\sin^{2}n\theta\pi$.
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Exercise XXIV, problem 9b, p. 122
$a\cos n\theta\pi + b\sin n\theta\pi$, $\sin^{2}n\theta\pi$, $a\cos^{2}n\theta\pi + b\sin^{2}n\theta\pi$.
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Exercise XXIV, problem 9c, p. 122
$a\cos n\theta\pi + b\sin n\theta\pi$, $\sin^{2}n\theta\pi$, $a\cos^{2}n\theta\pi + b\sin^{2}n\theta\pi$.
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Exercise XXV

Exercise XXV, problem 1, p. 124
If $\phi(n) \to +\infty$ and $\psi(n) \geq \phi(n)$ for all values of $n$, then $\psi(n) \to +\infty$.
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Exercise XXV, problem 10a, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n + (-1)^n > 10000pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 10000000pt minus 3pt(n n_0).
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Exercise XXV, problem 10b, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n + (-1)^n > 10000pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 10000000pt minus 3pt(n n_0).
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Exercise XXV, problem 11a, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 0pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 0pt minus 3pt(n n_0), $\Delta$ being any positive number.
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- [(*a*) $n_{0} = [\sqrtp{\Delta + 1}]$: (*b*) $n_{0} = 1 + [\Delta]$ or $2 + [\Delta]$, according as $[\Delta]$ is odd or even, *i.e.* $n_{0} = 1 + [\Delta] + \frac{1}{2} \{1 + (-1)^{[\Delta]}\}$.]
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Exercise XXV, problem 11b, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 0pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 0pt minus 3pt(n n_0), $\Delta$ being any positive number.
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- [(*a*) $n_{0} = [\sqrtp{\Delta + 1}]$: (*b*) $n_{0} = 1 + [\Delta]$ or $2 + [\Delta]$, according as $[\Delta]$ is odd or even, *i.e.* $n_{0} = 1 + [\Delta] + \frac{1}{2} \{1 + (-1)^{[\Delta]}\}$.]
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Exercise XXV, problem 12a, p. 124
Determine the least value of $n_{0}$ such that [1.5em][l](*a*) n/(n^2 + 1) < .0001, [1.5em][l](*b*) (1/n) + (-1)^n/n^2 < .00001, when $n \geq n_{0}$.
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Exercise XXV, problem 12b, p. 124
Determine the least value of $n_{0}$ such that [1.5em][l](*a*) n/(n^2 + 1) < .0001, [1.5em][l](*b*) (1/n) + (-1)^n/n^2 < .00001, when $n \geq n_{0}$.
Printed answer:
- [Let us take the latter case. In the first place (1/n) + (-1)^n/n^2 (n + 1)/n^2, and it is easy to see that the least value of $n_{0}$, such that $(n + 1)/n^{2} < .000\MS001$ when $n \geq n_{0}$, is $1\MC000\MC002$. But the inequality given is satisfied by $n = 1\MC000\MC001$, and this is the value of $n_{0}$ required.]
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Exercise XXV, problem 2, p. 124
If $\phi(n) \to 0$, and $|\psi(n)| \leq |\phi(n)|$ for all values of $n$, then $\psi(n) \to 0$.
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Exercise XXV, problem 3, p. 124
If $\lim |\phi(n)| = 0$, then $\lim \phi(n) = 0$.
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Exercise XXV, problem 4, p. 124
If $\phi(n)$ tends to a limit or oscillates finitely, and $|\psi(n)| \leq |\phi(n)|$ when $n \geq n_{0}$, then $\psi(n)$ tends to a limit or oscillates finitely.
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Exercise XXV, problem 5, p. 124
If $\phi(n)$ tends to $+\infty$, or to $-\infty$, or oscillates infinitely, and |(n)| |(n)| when $n \geq n_{0}$, then $\psi(n)$ tends to $+\infty$ or to $-\infty$ or oscillates infinitely.
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Exercise XXV, problem 6, p. 124
‘If $\phi(n)$ oscillates and, however great be $n_{0}$, we can find values of $n$ greater than $n_{0}$ for which $\psi(n) > \phi(n)$, and values of $n$ greater than $n_{0}$ for which $\psi(n) < \phi(n)$, then $\psi(n)$ oscillates’. Is this true? If not give an example to the contrary.
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Exercise XXV, problem 7, p. 124
If $\phi(n) \to l$ as $n \to \infty$, then also $\phi(n + p) \to l$, $p$ being any fixed integer. [This follows at once from the definition. Similarly we see that if $\phi(n)$ tends to $+\infty$ or $-\infty$ or oscillates so also does $\phi(n + p)$.]
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Exercise XXV, problem 8, p. 124
The same conclusions hold (except in the case of oscillation) if $p$ varies with $n$ but is always numerically less than a fixed positive integer $N$; or if $p$ varies with $n$ in any way, so long as it is always positive.
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Exercise XXV, problem 9a, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 9999990pt minus 3pt(n n_0), [1.5em][l](*b*) n^2 + 2n > 10000000pt minus 3pt(n n_0).
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Exercise XXV, problem 9b, p. 124
Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 9999990pt minus 3pt(n n_0), [1.5em][l](*b*) n^2 + 2n > 10000000pt minus 3pt(n n_0).
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Exercise XXVI

Exercise XXVI, problem 1, p. 131
What is the behaviour of the functions (n - 1n + 1)^2,0pt minus 3pt(-1)^n (n - 1n + 1)^2,0pt minus 3ptn^2 + 1n,0pt minus 3pt(-1)^n n^2 + 1n, as $n\to\infty$?
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Exercise XXVI, problem 2, p. 131
Which (if any) of the functions gather* 1/(^212n+ n^212n),0pt minus 3pt1/n(^212n+ n^212n), (n^212n+ ^212n)/ n(^212n+ n^212n) gather* tend to a limit as $n \to \infty$?
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Exercise XXVI, problem 3, p. 131
Denoting by $S(n)$ the general rational function of $n$ considered above, show that in all cases S(n + 1)S(n) = 1,0pt minus 3ptSn + (1/n)S(n) = 1.
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Exercise XXVII

Exercise XXVII, problem 1, p. 135
If $\phi(n)$ is positive and $\phi(n + 1) > K \phi(n)$, where $K > 1$, for all values of $n$, then $\phi(n) \to +\infty$.
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Exercise XXVII, problem 10, p. 135
Prove that if $x$ is positive then $\sqrt[n]{x} \to 1$ as $n \to \infty$.
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Exercise XXVII, problem 11, p. 135
$\sqrt[n]{n}\to 1$.
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Exercise XXVII, problem 12, p. 135
$\sqrtp[n]{n!} \to +\infty$.
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Exercise XXVII, problem 13, p. 135
Show that if $-1 < x < 1$ then u_n = m(m - 1) …(m - n + 1)n! x^n = mn x^n tends to zero as $n \to \infty$.
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Exercise XXVII, problem 2, p. 135
The same result is true if the conditions above stated are satisfied only when $n \geq n_{0}$.
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Exercise XXVII, problem 3, p. 135
If $\phi(n)$ is positive and $\phi(n + 1) < K\phi(n)$, where $0 < K < 1$, then $\lim\phi(n) = 0$. This result also is true if the conditions are satisfied only when $n \geq n_{0}$.
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Exercise XXVII, problem 4, p. 135
If $|\phi(n + 1)| < K|\phi(n)|$ when $n \geq n_{0}$, and $0 < K < 1$, then $\lim\phi(n) = 0$.
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Exercise XXVII, problem 5, p. 135
If $\phi(n)$ is positive and $\lim\{\phi(n + 1)\}/\{\phi(n)\} = l > 1$, then $\phi(n) \to +\infty$.
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Exercise XXVII, problem 6, p. 135
If $\lim\{\phi(n + 1)\}/\{\phi(n)\} = l$, where $l$ is numerically less than unity, then $\lim\phi(n) = 0$.
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Exercise XXVII, problem 7, p. 135
Determine the behaviour, as $n \to \infty$, of $\phi(n) = n^{r}x^{n}$, where $r$ is any positive integer.
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Exercise XXVII, problem 8, p. 135
Discuss $n^{-r}x^{n}$ in the same way.
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Exercise XXVII, problem 9, p. 135
Draw up a table to show how $n^{k}x^{n}$ behaves as $n \to \infty$, for all real values of $x$, and all positive and negative integral values of $k$.
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Exercise XXVIII

Exercise XXVIII, problem 1, p. 139
Verify (9) for $r = 2$, $3$, and (10) for $s = \frac{1}{2}$, $\frac{1}{3}$.
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Exercise XXVIII, problem 2, p. 139
Show that (9) and (10) are also true if $y > x > 0$.
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Exercise XXVIII, problem 3, p. 139
Show that (9) also holds for $r < 0$. [See Chrystal’s *Algebra*, vol. ii, pp. 43--45.]
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Exercise XXVIII, problem 4, p. 139
If $\phi(n) \to l$, where $l > 0$, as $n \to \infty$, then $\phi^{k} \to l^{k}$, $k$ being any rational number.
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Exercise XXVIII, problem 5, p. 139
Extend the results of xxvii. 7, 8, 9 to the case in which $r$ or $k$ are any rational numbers.
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Exercise XXXIII

Exercise XXXIII, problem 1, p. 157
Prove directly that $\phi(n) = r^{n} \cos n\theta$ converges to $0$ when $r < 1$ and to $1$ when $r = 1$ and $\theta$ is a multiple of $2\pi$. Prove further that if $r = 1$ and $\theta$ is not a multiple of $2\pi$, then $\phi(n)$ oscillates finitely; if $r > 1$ and $\theta$ is a multiple of $2\pi$, then $\phi(n) \to +\infty$; and if $r > 1$ and $\theta$ is not a multiple of $2\pi$, then $\phi(n)$ oscillates infinitely.
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Exercise XXXIII, problem 2, p. 157
Establish a similar series of results for $\phi(n) = r^{n} \sin n\theta$.
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Exercise XXXIII, problem 3, p. 157
Prove that gather* z^m + z^m+1 + …= z^m/(1 - z), z^m + 2z^m+1 + 2z^m+2 + …= z^m(1 + z)/(1 - z), gather* if and only if $|z| < 1$. Which of the theorems of [§]86 do you use?
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Exercise XXXIII, problem 4, p. 157
Prove that if $-1 < r < 1$ then 1 + 2r+ 2r^22+ … = (1 - r^2)/(1 - 2r+ r^2).
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Exercise XXXIII, problem 5, p. 157
The series 1 + z1 + z + (z1 + z)^2 + … converges to the sum $1\bigg/\left(1 - \dfrac{z}{1 + z}\right) = 1 + z$ if $|z/(1 + z) | < 1$. Show that this condition is equivalent to the condition that $z$ has a real part greater than $-\frac{1}{2}$.
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Exercise Misc-IV

Exercise Misc-IV, problem 1, p. 157
The function $\phi(n)$ takes the values $1$, $0$, $0$, $0$, $1$, $0$, $0$, $0$, $1$, … when $n = 0$, $1$, $2$, …. Express $\phi(n)$ in terms of $n$ by a formula which does not involve trigonometrical functions.
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Exercise Misc-IV, problem 2, p. 157
If $\phi(n)$ steadily increases, and $\psi(n)$ steadily decreases, as $n$ tends to $\infty$, and if $\psi(n) > \phi(n)$ for all values of $n$, then both $\phi(n)$ and $\psi(n)$ tend to limits, and $\lim\phi(n) \leq \lim\psi(n)$.
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Exercise Misc-IV, problem 3, p. 157
Prove that, if (n) = (1 + 1n)^n,0pt minus 3pt(n) = (1 - 1n)^-n, then $\phi(n + 1) > \phi(n)$ and $\psi(n + 1) < \psi(n)$.
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