THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE

Excerpts

THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The logarithm of $x$ tends to infinity with $x$, but more slowly than **** positive power of $x$, integral or fractional.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Since $\log x$ is an increasing function of $x$, in the stricter sense of [§]95, it can only pass once through the value $1$. Hence our definition does in fact define one definite number.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The series on the right-hand side of this equation is known as the **series**.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Another very important expansion in powers of $x$ is that for $\log(1 + x)$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The value of $\gamma$ is in fact $.577\dots$, and $\gamma$ is usually called **’s constant**.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The power series for $e^{x}$ is so important that it is worth while to investigate it by an alternative method which does not depend upon Taylor’s Theorem.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
If $x$ lies outside these limits the series is not convergent.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We know that this series is convergent for all values of $x$, and we may therefore define the function $\exp x$ by the equation
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We define $\log x$, the logarithm of $x$, by the equation x = _1^x dtt.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Perhaps the most interesting feature of the function $\log x$ is its behaviour as $x$ tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. *The logarithm of $x$ tends to infinity with $x$, but more slowly than **** positive power of $x$, integral or fractional.*
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
This fact is sometimes expressed loosely by saying that the ‘order of infinity of $\log x$ is infinitely small’; but the reader will hardly require at this stage to be warned against such modes of expression.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We define $e$ as *the number whose logarithm is $1$*. In other words $e$ is defined by the equation 1 = _1^e dtt.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We now define the *exponential function* $e^{y}$ for all real values of $y$ as the inverse of the logarithmic function. In other words we write x = e^y if $y = \log x$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Thus *the derivative of the exponential function is equal to the function itself*. More generally, if $x = e^{ay}$ then $dx/dy = ae^{ay}$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We take this as our *definition* of $a^{x}$ when $x$ is irrational. Thus $10^{\sqrt{2}} = e^{\sqrt{2}\log 10}$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We saw however in [§]200 that with the aid of logarithms we can construct functions which tend to zero, as $n \to \infty$, more rapidly than $1/n$, yet less rapidly than $n^{-1-\alpha}$, however small $\alpha$ may be, provided of course that it is positive.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The reader should observe the extreme rapidity with which the higher exponential functions, such as $e^{e^{x}}$ and $e^{e^{e^{x}}}$, increase with $x$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Conversely, the rate of increase of the higher logarithmic functions is extremely slow. Thus to make $\log\log\log\log x > 1$ we have to suppose $x$ a number with over $8000$ figures.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The reader will observe that the exponential series has the property of reproducing itself when every term is differentiated, and that no other series of powers of $x$ would possess this property: for some further remarks in this connection see AppendixII.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Verify that these formulae may be deduced from the corresponding formulae in $\cos x$ and $\sin x$, by writing $\cosh x$ for $\cos x$ and $i\sinh x$ for $\sin x$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We require to show that the limit of $R_{m}$, when $m$ tends to $\infty$, is zero.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
The only difference is that the proof is a little simpler; for, since $\arctan x$ is an odd function of $x$, we need only consider positive values of $x$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
If $x = 1$, we obtain the formula 14= 1 - 13 + 15 - ….
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
This series may be used to calculate $\log 2$, a purpose for which the series $1 - \frac{1}{2} + \frac{1}{3} - \dots$, owing to the slowness of its convergence, is practically useless.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Let us consider the error committed in taking $8\frac{3}{16}$ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than $3^{2}/64^{2}$, which is less than $.003$.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
We shall now give an outline of a method of investigation of the properties of $e^{x}$ and $\log x$ entirely different in logical order from that followed in the preceding pages.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
Incidentally we have proved that $\exp x$ is a continuous function.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
If $n$ is not divisible by $10$, and $\log_{10}n = p/q$, we have $10^{p} = n^{q}$, which is impossible, since $10^{p}$ ends with $0$ and $n^{q}$ does not.

Equations

THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x = \int \frac{dx}{x}
The logarithm of x is defined as the integral of 1/x, as already introduced in Chapter VI.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x = \int_{1}^{x} \frac{dt}{t}
Definition of log x as the integral of dt/t from 1 to x, for positive x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x} \log x = 1/x
The derivative of log x with respect to x is 1/x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x = \int_{1}^{x} \frac{dt}{t} = -\int_{x}^{1} \frac{dt}{t} < 0
For 0 < x < 1, log x is negative, since reversing the limits of the integral changes its sign.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x = \int_{1}^{x} \frac{dt}{t} = -\int_{1}^{1/x} \frac{du}{u} = -\log(1/x)
Substituting t = 1/u shows that log x equals minus log(1/x).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
f(xy) = f(x) + f(y)
The logarithm satisfies the functional equation: the function of a product is the sum of the functions of the factors.
This equation is in DERIVATIVES AND INTEGRALS (DERIVATIVES AND INTEGRALS)
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x^{n} = n\log x
The logarithm of a positive integer power of x is n times log x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log e^{n} = n\log e = n
The logarithm of e to a positive integer power n equals n, since log e = 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log e^{y} = y
The logarithm of e raised to y is y, for rational y (extended later to all real y).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
y = \log x,\quad x = e^{y}
The equations y = log x and x = e^y are consequences of one another.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
1 = \int_{1}^{e} \frac{dt}{t}
Definition of the number e as the number whose logarithm is 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
dy/dx = 1/x
If x = e^y, the derivative of y = log x with respect to x is 1/x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\frac{dx}{dy} = x = e^{y}
The derivative of the exponential function equals the function itself.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
dx/dy = ae^{ay}
If x = e^{ay} then the derivative of x with respect to y is a e^{ay}.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
f(y + z) = f(y)f(z)
The exponential function satisfies the functional equation that turns addition of exponents into multiplication.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
e^{-y} = 1/e^{y}
A negative exponent gives the reciprocal of the positive power.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\lim y^{\alpha}/e^{y} = \lim e^{-y}y^{\alpha} = 0
e^y tends to infinity faster than any power of y, so y^alpha / e^y tends to zero as y tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\frac{\log x}{x^{\alpha}} \to 0
log x tends to infinity more slowly than any positive power of x, as x tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\lim_{y\to +0} y^{\alpha} \log y = -\lim_{x\to +\infty} (\log x)/x^{\alpha} = 0
As x tends to zero through positive values, log x tends to minus infinity, more slowly than any power of 1/x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log x < (x^{\beta} - 1)/\beta < x^{\beta}/\beta
For x > 1 and any positive beta, log x is bounded above by (x^beta - 1)/beta, which is less than x^beta/beta.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(\log\log y)/(\log y)^{\alpha} = (\log x)/x^{\alpha} \to 0
log log y tends to infinity more slowly than any power of log y as y tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(\log x)/x^{\alpha} = -y^{\alpha} \log y
With x = 1/y, the quotient log x over x^alpha equals minus y^alpha log y.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
x = a^{y},\quad y = \log_{a} x
The logarithm of x to base a is the exponent y such that a^y = x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log_{10} x = (\log_{e} x)/(\log_{e} 10)
The common logarithm of x equals the natural logarithm of x divided by the natural logarithm of 10.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(a^{x})^{y} = a^{xy}
The power of a power law a^(xy) holds for all real exponents.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
y = e^{x\log a}
Definition of the general power a^x for irrational x as e^(x log a).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log a^{x} = x\log a
The logarithm of a^x equals x times log a.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
a^{x} = e^{x\log a} = e^{\alpha x}
For a > 1, a^x equals e^(alpha x) with alpha positive, so a^x tends to infinity as x tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
a^{x} = e^{x\log a} = e^{-\beta x}
For a < 1, a^x equals e^(-beta x) with beta positive, so a^x tends to zero as x tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x} e^{x\log a} = e^{x\log a} \log a = a^{x} \log a
The derivative of a^x with respect to x is a^x times log a.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{a} e^{x\log a} = e^{x\log a} (x/a) = xa^{x-1}
The derivative of a^x with respect to the base a is x a^(x-1).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(a^{x} - 1)/x \to \log a
As x tends to zero, (a^x - 1)/x tends to log a.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^{n} = \lim_{n\to\infty} \left(1 - \frac{x}{n}\right)^{-n} = e^{x}
The exponential e^x is the limit of (1 + x/n)^n and of (1 - x/n)^(-n) as n tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\lim_{\xi\to\infty} \left(1 + \frac{x}{\xi}\right)^{\xi} = \lim_{\xi\to -\infty} \left(1 + \frac{x}{\xi}\right)^{\xi} = e^{x}
The generalisation of the limit representation of e^x to a continuous variable xi tending to plus or minus infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
n(1 - x^{-1/n}) < \log x < n(x^{1/n} - 1)
For x > 1 and any positive integer n, log x lies between n(1 - x^(-1/n)) and n(x^(1/n) - 1).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\left(1 + \frac{y}{n}\right)^{n} < x < \left(1 - \frac{y}{n}\right)^{-n}
With y = log x and x = e^y, x lies between (1 + y/n)^n and (1 - y/n)^(-n).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\lim n(1 - x^{-1/n}) = \lim n(x^{1/n} - 1) = \log x
The logarithm of x is the limit of n(x^(1/n) - 1) and of n(1 - x^(-1/n)) as n tends to infinity.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x}(\log x)^{1-s} = \frac{1 - s}{x(\log x)^{s}}
The derivative of (log x) raised to the power 1 − s equals (1 − s) divided by x(log x)^s.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x}\log\log x = \frac{1}{x\log x}
The derivative of log log x with respect to x equals 1 divided by x log x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\int_{a}^{\xi} \frac{dx}{x(\log x)^{s}} = \frac{(\log\xi)^{1-s} - (\log a)^{1-s}}{1 - s}
The definite integral of 1/(x(log x)^s) between a and ξ equals the difference of the powers of logarithms over 1 − s.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\int_{\DPtypo{}{a}}^{\xi} \frac{dx}{x\log x} = \log\log \xi - \log\log a
The definite integral of 1/(x log x) between a and ξ equals log log ξ minus log log a.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\sum_{n_{0}}^{\infty} \frac{1}{n(\log n)^{s}}
The series of 1/(n(log n)^s) converges if s > 1 and diverges if s ≤ 1, with the matching integral test.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{n-1}}{(n - 1)!} + \frac{x^{n}}{n!} e^{\theta x}
Taylor's theorem expansion of e^x with remainder term, where 0 < θ < 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
x^{n}/n! \to 0
x^n divided by n! tends to zero as n tends to infinity, whatever the value of x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{n}}{n!} + \dots
e^x equals the infinite power series 1 + x + x²/2! + … ; this is the exponential series.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
e = 1 + 1 + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots
Euler's number e is the sum of the series 1 + 1 + 1/2! + 1/3! + …
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\left(1 + 1 + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots\right)^{x} = 1 + x + \frac{x^{2}}{2!} + \dots + \frac{x^{n}}{n!} + \dots
The x-th power of the series for e equals the exponential series in x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
a^{x} = e^{x\log a} = 1 + (x\log a) + \frac{(x\log a)^{2}}{2!} + \dots
a^x equals e^(x log a), and so equals the exponential series in x log a, for positive a.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\left(1 + \frac{x}{n}\right)^{n} < E_{n}(x) < \left(1 - \frac{x}{n}\right)^{-n}
For x > 0 and n > x, the partial sum E_n(x) of the exponential series lies between (1 + x/n)^n and (1 − x/n)^(−n).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
f(x)f(y) = f(x + y)
The exponential series satisfies the functional equation f(x)f(y) = f(x+y).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
f(x)f(-x) = f(0) = 1
Since f(x)f(y) = f(x+y), f(x) times f(−x) equals f(0), which is 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log(1 + x) = \int_{0}^{x} \frac{dt}{1 + t}
log(1 + x) is the definite integral of 1/(1 + t) from 0 to x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
1/(1 + t) = 1 - t + t^{2} - \dots + (-1)^{m-1} t^{m-1} + \frac{(-1)^{m} t^{m}}{1 + t}
1/(1 + t) equals the first m terms of the geometric-type series 1 − t + t² − … plus a remainder term.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
R_{m} = \int_{0}^{x} \frac{t^{m}\, dt}{1 + t}
The remainder R_m is the integral of t^m/(1 + t) from 0 to x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
R_{m} = (-1)^{m} \int_{0}^{\xi} \frac{u^{m}\, du}{1 - u}
For negative x, with x = −ξ and t = −u, the remainder R_m equals (−1)^m times an integral with positive integrand.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
0 < |R_{m}| < \frac{1}{1 - \xi} \int_{0}^{\xi} u^{m}\, du
The absolute value of R_m is positive and bounded by the integral of u^m scaled by 1/(1 − ξ), which tends to zero.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log(1 + x) = x - \tfrac{1}{2} x^{2} + \tfrac{1}{3} x^{3} - \dots
log(1 + x) equals x − x²/2 + x³/3 − … for −1 < x ≤ 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\log 2 = 1 - \tfrac{1}{2} + \tfrac{1}{3} - \dots
The alternating harmonic series 1 − 1/2 + 1/3 − … sums to log 2, the case x = 1 of the logarithmic series.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\arctan x = \int_{0}^{x} \frac{dt}{1 + t^{2}}
The inverse tangent of x is the integral of dt/(1 + t^2) from 0 to x.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\tfrac{1}{4}\pi = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \dots
Setting x = 1 in the arctangent series gives pi/4 as the alternating series 1 - 1/3 + 1/5 - ...
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\argtanh x = \frac{1}{2} \log\left(\dfrac{1 + x}{1 - x}\right)
The inverse hyperbolic tangent of x equals half the logarithm of (1 + x)/(1 - x).
This equation is in ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\ INTEGRAL CALCULUS (ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\ INTEGRAL CALCULUS)
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(1 + x)^{m} = e^{m\log(1+ x)}
For irrational m, (1 + x)^m is written as e to the power m log(1 + x).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x}(1 + x)^{m} = \{m/(1 + x)\} e^{m\log(1 + x)} = m(1 + x)^{m-1}
The derivative of (1 + x)^m with respect to x is m(1 + x)^(m-1), so the differentiation rule is unchanged for irrational m.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\exp x = 1 + x + \frac{x^{2}}{2!} + \dots
The function exp x is defined as the sum of the exponential series.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\exp x × \exp y = \exp(x + y)
The exponential of a sum is the product of the exponentials.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
D_{x} \exp x = \exp x
The derivative of exp x with respect to x is exp x itself.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\frac{dy}{dx} = y
With y = exp x, the derivative dy/dx equals y.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
x = \int_{1}^{y} \frac{dt}{t}
The logarithm of y is the integral of dt/t from 1 to y, so x is the logarithm of y.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
(\exp x)^{n} = \exp nx
For a positive integer n, the n-th power of exp x equals exp(nx).
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\exp x \exp(-x) = 1
The product of exp x and exp(-x) is 1, which extends the result to negative values.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
\exp x = (\exp 1)^{x} = e^{x}
exp x is the same as e raised to the power x, where e is exp 1.
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE
e = \exp 1 = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \dots
Euler's number e is exp 1, given by the sum of the exponential series at x = 1.

Problems

Exercise LXXXII

Exercise LXXXII, problem 1, p. 359
Prove from the definition that if $u > 0$ then u/(1 + u) < (1 + u) < u.
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Exercise LXXXII, problem 2, p. 359
Prove that $\log(1 + u)$ lies between $u - \dfrac{u^{2}}{2}$ and $u - \dfrac{u^{2}}{2(1 + u)}$ when $u$ is positive.
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Exercise LXXXII, problem 3, p. 359
If $0 < u < 1$ then $u < -\log(1 - u) < u/(1 - u)$.
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Exercise LXXXII, problem 4, p. 359
Prove that _x1 xx - 1 = _t0 (1 + t)t = 1.
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Exercise LXXXVII

The data holds no problems for this exercise yet.

Exercise LXXXVIII

Exercise LXXXVIII, problem 1, p. 375
The series 1n(n)^2,0pt minus 3pt(n)^100n^101/100,0pt minus 3ptn^2 - 1n^2 + 1 1n(n)^7/6 are convergent. [The convergence of the first series is a direct consequence of the theorem of the preceding section. That of the second follows from the fact that $(\log n)^{100}$ is less than $n^{\beta}$ for sufficiently large values of $n$, however small $\beta$ may be, provided that it is positive. And so, taking $\beta = 1/200$, $(\log n)^{100} n^{-101/100}$ is less than $n^{-201/200}$ for sufficiently large values of $n$. The convergence of the third series follows from the comparison test at the end of the last section.]
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Exercise LXXXVIII, problem 2, p. 375
The series 1n(n)^6/7,0pt minus 3pt1n^100/101(n)^100,0pt minus 3ptnn(nn)^2 + 1 are divergent.
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Exercise LXXXVIII, problem 3, p. 375
The series (n)^pn^1+s,0pt minus 3pt(n)^p (n)^qn^1+s,0pt minus 3pt(n)^pn(n)^1+s, where $s > 0$, are convergent for all values of $p$ and $q$; similarly the series 1n^1-s(n)^p,0pt minus 3pt1n^1-s(n)^p(n)^q,0pt minus 3pt1n(n)^1-s(n)^p are divergent.
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Exercise LXXXVIII, problem 4, p. 375
The question of the convergence or divergence of such series as 1nnn,0pt minus 3ptnnnn cannot be settled by the theorem of p.375, since in each case the function under the sign of summation tends to zero more rapidly than $1/(n\log n)$ yet less rapidly than $n^{-1}(\log n)^{-1-\alpha}$, where $\alpha$ is any positive number however small. For such series we need a still more delicate test. The reader should be able, starting from the equations align* D_x(_kx)^1-s &= 1 - sx x _2x …_k-1 x (_kx)^s, D_x_k+1x &= 1x x _2x …_k-1x _kx, align* where $\log_{2}x = \log\log x$, $\log_{3} x = \log\log\log x$, …, to prove the following theorem: *the series and integral _n_0^ 1n n _2n …_k-1n (_kn)^s,0pt minus 3pt_a^ dxx x _2x …_k-1x (_kx)^s are convergent if $s > 1$ and divergent if $s \leq 1$*, 0.375em plus 0.75em minus 0.25em$n_{0}$ and $a$ being any numbers sufficiently great to ensure that $\log_{k}n$ and $\log_{k}x$ are positive when $n \geq n_{0}$ or $x \geq a$. These values of $n_{0}$ and $a$ increase very rapidly as $k$ increases: thus $\log x > 0$ requires $x > 1$, $\log_{2}x > 0$ requires $x > e$, $\DPtypo{\log\log x}{\log_{3}x} > 0$ requires $x > e^{e}$, and so on; and it is easy to see that $e^{e} > 10$, $e^{e^{e}} > e^{10} > 20,000$, $e^{e^{e^{e}}} > e^{20,000} > 10^{8000}$.
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Exercise LXXXVIII, problem 5, p. 375
Prove that the integral $\ds\int_{0}^{a} \frac{1}{x} \left\{\log \left(\frac{1}{x}\right)\right\}^{s} dx$, where $0 < a < 1$, is convergent if $s < -1$, divergent if $s \geq -1$. [Consider the behaviour of _^a 1x (1x)^s dx as $\epsilon \to +0$. This result also may be refined upon by the introduction of higher logarithmic factors.]
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Exercise LXXXVIII, problem 6, p. 375
Prove that $\ds\int_{0}^{1} \frac{1}{x} \left\{\log \left(\frac{1}{x}\right)\right\}^{s} dx$ has no meaning for any value of $s$. [The last example shows that $s < -1$ is a necessary condition for convergence at the lower limit: but $\{\log(1/x)\}^{s}$ tends to $\infty$ like $(1 - x)^{s}$, as $x \to 1 - 0$, if $s$ is negative, and so the integral diverges at the upper limit when $s < -1$.]
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Exercise LXXXVIII, problem 7, p. 375
0.375em plus 0.75em minus 0.25emThe necessary and sufficient conditions for the convergence of $\ds\int_{0}^{1} x^{a-1} \left\{\log \left(\frac{1}{x}\right)\right\}^{s} dx$ are $a > 0$, $s > -1$.
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Exercise LXXXIX

Exercise LXXXIX, problem 1, p. 377
**’s limit.** Show that (n) = 1 + 12 + 13 + …+ 1n - 1 - n tends to a limit $\gamma$ as $n \to \infty$, and that $0 < \gamma \leq 1$. [This follows at once from [§]174. The value of $\gamma$ is in fact $.577\dots$, and $\gamma$ is usually called **’s constant**.]
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Exercise LXXXIX, problem 2, p. 377
If $a$ and $b$ are positive then 1a + 1a + b + 1a + 2b + … + 1a + (n - 1) b - 1b(a + nb tends to a limit as $n \to \infty$.
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Exercise LXXXIX, problem 3, p. 377
If $0 < s < 1$ then (n) = 1 + 2^-s + 3^-s + …+ (n - 1)^-s - n^1-s1 - s tends to a limit as $n \to \infty$.
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Exercise LXXXIX, problem 4, p. 377
Show that the series 11 + 12(1 + 12) + 13(1 + 12 + 13) + … is divergent. [Compare the general term of the series with $1/(n\log n)$.] Show also that the series derived from $\sum n^{-s}$, in the same way that the above series is derived from $\sum (1/n)$, is convergent if $s > 1$ and otherwise divergent.
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Exercise LXXXIX, problem 5, p. 377
Prove generally that if $\sum u_{n}$ is a series of positive terms, and s_n = u_1 + u_2 + …+ u_n, then $\sum (u_{n}/s_{n-1})$ is convergent or divergent according as $\sum u_{n}$ is convergent or [pg]378 divergent. [If $\sum u_{n}$ is convergent then $s_{n-1}$ tends to a positive limit $l$, and so $\sum (u_{n}/s_{n-1})$ is convergent. If $\sum u_{n}$ is divergent then $s_{n-1} \to \infty$, and u_n/s_n-1 > 1 + (u_n/s_n-1) = (s_n/s_n-1) (% [examples:lxxxii]Ex. lxxxii%. 1); and it is evident that (s_2/s_1) + (s_3/s_2) + …+ (s_n/s_n-1) = (s_n/s_1) tends to $\infty$ as $n \to \infty$.]
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Exercise LXXXIX, problem 6, p. 377
Prove that the same result holds for the series $\sum (u_{n}/s_{n})$. [The proof is the same in the case of convergence. If $\sum u_{n}$ is divergent, and $u_{n} < s_{n-1}$ from a certain value of $n$ onwards, then $s_{n} < 2s_{n-1}$, and the divergence of $\sum (u_{n}/s_{n})$ follows from that of $\sum (u_{n}/s_{n-1})$. If on the other hand $u_{n} \geq s_{n-1}$ for an infinity of values of $n$, as might happen with a rapidly divergent series, then $u_{n}/s_{n} \geq \frac{1}{2}$ for all these values of $n$.]
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Exercise LXXXIX, problem 7, p. 377
Sum the series $1 - \frac{1}{2} + \frac{1}{3} - \dots$. [We have 1 + 12 + …+ 12n = (2n + 1) + + _n, 0pt minus 3pt2(12 + 14 + …+ 12n) = (n + 1) + + _n’, by Ex. 1, $\gamma$ denoting Euler’s constant, and $\epsilon_{n}$, $\epsilon_{n}'$ being numbers which tend to zero as $n \to \infty$. Subtracting and making $n \to \infty$ we see that the sum of the given series is $\log 2$. See also [§]213.]
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Exercise LXXXIX, problem 8, p. 377
Prove that the series _0^ (-1)^n(1 + 12 + …+ 1n + 1 - n - C) oscillates finitely except when $C = \gamma$, when it converges.
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Exercise XC

Exercise XC, problem 1, p. 379
Show that x = 1 + x^22! + x^44! + …,0pt minus 3ptx = x + x^33! + x^55! + ….
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Exercise XC, problem 10, p. 379
Prove that $\sum\limits_{1}^{\infty} \dfrac{(n - 1)x^{n}}{(n + 2)n!} = \left\{(x^{2} - 3x + 3)e^{x} + \frac{1}{2}x^{2} - 3\right\}/x^{2}$. [Multiply numerator and denominator by $n + 1$, and proceed as in Ex. 7.]
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Exercise XC, problem 11, p. 379
Determine $a$, $b$, $c$ so that $\{(x + a)e^{x} + (bx + c)\}/x^{3}$ tends to a limit as $x \to 0$, evaluate the limit, and draw the graph of the function $e^{x} + \dfrac{bx + c}{x + a}$.
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Exercise XC, problem 12, p. 379
Draw the graphs of $1 + x$, $1 + x + \frac{1}{2}x^{2}$, $1 + x + \frac{1}{2}x^{2} + \frac{1}{6}x^{3}$, and compare them with that of $e^{x}$.
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Exercise XC, problem 13, p. 379
Prove that $e^{-x} - 1 + x - \dfrac{x^{n}}{2!} + \dots - (-1)^{n}\dfrac{x^{n}}{n!}$ is positive or negative according as $n$ is odd or even. Deduce the exponential theorem.
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Exercise XC, problem 14, p. 379
If X_0 = e^x,0pt minus 3ptX_1 = e^x - 1,0pt minus 3ptX_2 = e^x - 1 - x,0pt minus 3ptX_3 = e^x - 1 - x - (x^2/2!), …, then $dX_{\nu}/dx = X_{\nu-1}$. Hence prove that if $t > 0$ then X_1(t) = _0^t X_0 dx < te^t,0pt minus 3ptX_2(t) = _0^t X_1 dx < _0^t xe^x dx < e^t _0^t x dx = t^22! e^t, and generally $X_{\nu}(t) < \dfrac{t^{\nu}}{\nu!} e^{t}$. Deduce the exponential theorem.
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Exercise XC, problem 15, p. 379
Show that the expansion in powers of $p$ of the positive root of $x^{2+p} = a^{2}$ begins with the terms a1 - 12 pa + 18 p^2a (2 + a). % [0]% (*Math. Trip.* 1909.)% [1]%
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Exercise XC, problem 2, p. 379
If $x$ is positive then the greatest term in the exponential series is the $([x] + 1)$-th, unless $x$ is an integer, when the preceding term is equal to it.
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Exercise XC, problem 3, p. 379
Show that $n! > (n/e)^{n}$. [For $n^{n}/n!$ is one term in the series for $e^{n}$.]
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Exercise XC, problem 4, p. 379
Prove that $e^{n} = (n^{n}/n!)(2 + S_{1} + S_{2})$, where S_1 = 11 + + 1(1 + )(1 + 2) + …,0pt minus 3ptS_2 = (1 - ) + (1 - )(1 - 2) + …, and $\nu = 1/n$; and deduce that $n!$ lies between $2(n/e)^{n}$ and $2(n + 1)(n/e)^{n}$.
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Exercise XC, problem 5, p. 379
Employ the exponential series to prove that $e^{x}$ tends to infinity more rapidly than any power of $x$. [Use the inequality $e^{x} > x^{n}/n!$.]
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Exercise XC, problem 6, p. 379
Show that $e$ is not a rational number. [If $e = p/q$, where $p$ and $q$ are integers, we must have pq = 1 + 12!+13! + …+ 1q! + … or, multiplying up by $q!$, q! (pq - 1 - 1 - 12! - …- 1q!) = 1q + 1 + 1(q + 1)(q + 2) + … and this is absurd, since the left-hand side is integral, and the right-hand side less than $\{1/(q + 1)\} + \{1/(q + 1)\}^{2} + \dots = 1/q$.]
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Exercise XC, problem 7, p. 379
Sum the series $\sum\limits_{0}^{\infty} P_{r}(n)\dfrac{x^{n}}{n!}$, where $P_{r}(n)$ is a polynomial of degree $r$ in $n$. [We can express $P_{r}(n)$ in the form A_0 + A_1n + A_2n(n - 1) + …+ A_rn(n - 1) …(n - r + 1), and align* _0^ P_r(n) x^nn! &= A_0_0^x^nn! + A_1_1^x^n(n - 1)! + … + A_r_r^x^n(n - r)! &= (A_0 + A_1x + A_2x^2 + …+ A_rx^r)e^x.] align*
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Exercise XC, problem 8, p. 379
Show that _1^ n^3n! x^n = (x + 3x^2 + x^3)e^x,0pt minus 3pt_1^ n^4n! x^n = (x + 7x^2 + 6x^3 + x^4)e^x; and that if $S_{n} = 1^{3} + 2^{3} + \dots + n^{3}$ then _1^ S_nx^nn! = 14(4x + 14x^2 + 8x^3 + x^4)e^x. In particular the last series is equal to zero when $x = -2$. % [0]% (*Math. Trip.* 1904.)% [1]%
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Exercise XC, problem 9, p. 379
Prove that $\sum (n/n!) = e$, $\sum (n^{2}/n!) = 2e$, $\sum (n^{3}/n!) = 5e$, and that $\sum (n^{k}/n!)$, where $k$ is any positive integer, is a positive integral multiple of $e$.
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Exercise XCI

Exercise XCI, problem 1, p. 382
$\log \left(\dfrac{1}{1 - x}\right) = x + \frac{1}{2} x^{2} + \frac{1}{3} x^{3} + \dots$ if $-1 \leq x < 1$.
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Exercise XCI, problem 10, p. 382
Show that 14= (1/2) + (1/3) = 4(1/5) - (1/239), and calculate $\pi$ to $6$ places of decimals.
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Exercise XCI, problem 11, p. 382
Show that the expansion of $(1 + x)^{1+x}$ in powers of $x$ begins with the terms $1 + x + x^{2} + 1/2 x^{3}$.
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Exercise XCI, problem 12, p. 382
Show that _10 e - x(x + 1) _10(1 + xx) = _10 e24x^2, approximately, for large values of $x$. Apply the formula, when $x = 10$, to obtain an approximate value of $\log_{10} e$, and estimate the accuracy of the result.
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Exercise XCI, problem 13, p. 382
Show that 11 - x (11 - x) = x + (1 + 12)x^2 + (1 + 12 + 13)x^3 + …, if $-1 < x < 1$.
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Exercise XCI, problem 14, p. 382
0.375em plus 0.75em minus 0.25emUsing the logarithmic series and the facts that $\log_{10} 2.3758 = .375\MS809\MS9\dots$ and $\log_{10} e = .4343\dots$, show that an approximate solution of the equation $x = 100 \log_{10}x$ is $237.581\MS21$.
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Exercise XCI, problem 15, p. 382
Expand $\log\cos x$ and $\log(\sin x/x)$ in powers of $x$ as far as $x^{4}$, and verify that, to this order, x = x - 145 x + 644512x.
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Exercise XCI, problem 16, p. 382
Show that %[** TN: In-line in the original] _0^x dt1 + t^4 = x - 15x^5 + 19x^9 - … if $-1 \leq x \leq 1$. Deduce that 1 - 15 + 19 - … = + 2(2 + 1)/42.
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Exercise XCI, problem 17, p. 382
Prove similarly that 13 - 17 + 111 - … = _0^1 t^2 dt1 + t^4 = - 2(2 + 1)/42.
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Exercise XCI, problem 18, p. 382
Prove generally that if $a$ and $b$ are positive integers then 1a - 1a + b + 1a + 2b - … = _0^1 t^a-1 dt1 + t^b, and so that the sum of the series can be found. Calculate in this way the sums of $1 - \frac{1}{4} + \frac{1}{7} - \dots$ and $\frac{1}{2} - \frac{1}{5} + \frac{1}{8} - \dots$.
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Exercise XCI, problem 2, p. 382
$\argtanh x = \frac{1}{2} \log\left(\dfrac{1 + x}{1 - x}\right) = x + \frac{1}{3} x^{3} + \frac{1}{5} x^{5} + \dots$ if $-1 < x < 1$.
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Exercise XCI, problem 3, p. 382
Prove that if $x$ is positive then (1 + x) = x1 + x + 12 (x1 + x)^2 + 13 (x1 + x)^3 + ….
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Exercise XCI, problem 4, p. 382
Obtain the series for $\log(1 + x)$ and $\arctan x$ by means of Taylor’s theorem.
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Exercise XCI, problem 5, p. 382
If $y > 0$ then y = 2 y - 1y + 1 + 13 (y - 1y + 1)^3 + 15 (y - 1y + 1)^5 + ….
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Exercise XCI, problem 6, p. 382
Find $\log 10$ to $3$ places of decimals from the formula 10 = 32 + (1 + 14).
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Exercise XCI, problem 7, p. 382
Prove that (x + 1x) = 212x + 1 + 13(2x + 1)^3 + 15(2x + 1)^5 + … if $x > 0$, and that (x - 1)^2(x + 2)(x + 1)^2(x - 2) = 22x^3 - 3x + 13(2x^3 - 3x)^3 + 15(2x^3 - 3x)^5 + … if $x > 2$. Given that $\log 2 = .693\MS147\MS1\dots$ and $\log 3 = 1.098\MS612\MS3\dots$, show, by putting $x = 10$ in the second formula, that $\log 11 = 2.397\MS895\dots$.
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Exercise XCI, problem 8, p. 382
Show that if $\log 2$, $\log 5$, and $\log 11$ are known, then the formula 13 = 311 + 5 - 92 gives $\log 13$ with an error practically equal to $.000\MS15$.
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Exercise XCI, problem 9, p. 382
Show that 12 2 = 7a + 5b + 3c,0pt minus 3pt12 3 = 11a + 8b + 5c,0pt minus 3pt12 5 = 16a + 12b + 7c, where $a = \argtanh(1/31)$, $b = \argtanh(1/49)$, $c = \argtanh(1/161)$.
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Exercise XCII

Exercise XCII, problem 1, p. 385
Prove that if $-1 < x < 1$ then 11 + x^2 = 1 - 12x^2 + 1·32·4x^4 - …,0pt minus 3pt11 - x^2 = 1 + 12x^2 + 1·32·4x^4 + ….
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Exercise XCII, problem 2, p. 385
**to quadratic and other surds.** 0.375em plus 0.75em minus 0.25emLet $\sqrt{M}$ be a quadratic surd whose numerical value is required. Let $N^{2}$ be the square nearest to $M$; and let $M = N^{2} + x$ or $M = N^{2} - x$, $x$ being positive. Since $x$ cannot be greater than $N$, $x/N^{2}$ is comparatively small and the surd $\sqrt{M} = N\sqrtb{1 ± (x/N^{2})}$ can be expressed in a series = N 1 ± 12(xN^2) - 1·12·4(xN^2)^2 ± …, which is at any rate fairly rapidly convergent, and may be very rapidly so. Thus 67 = 64 + 3 = 8 1 + 12(364) - 1·12·4(364)^2 + …. Let us consider the error committed in taking $8\frac{3}{16}$ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than $3^{2}/64^{2}$, which is less than $.003$.
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Exercise XCII, problem 3, p. 385
If $x$ is small compared with $N^{2}$ then N^2 + x = N + x4N + Nx2(2N^2 + x), the error being of the order $x^{4}/N^{7}$. Apply the process to $\sqrt{907}$. [Expanding by the binomial theorem, we have N^2 + x = N + x2N - x^28N^3 + x^316N^5, the error being less than the numerical value of the next term, viz. $5x^{4}/128N^{7}$. Also Nx2(2N^2 + x) = x4N (1 + x2N^2)^-1 = x4N - x^28N^3 + x^316N^5, the error being less than $x^{4}/32N^{7}$. The result follows. The same method may be applied to surds other than quadratic surds, *e.g.* to $\sqrt[3]{1031}$.]
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Exercise XCII, problem 4, p. 385
If $M$ differs from $N^{3}$ by less than $1$ per cent. of either then $\sqrt[3]{M}$ differs from $\frac{2}{3}N + \frac{1}{3}(M/N^{2})$ by less than $N/90\MC000$. % [0]% (*Math. Trip.* 1882.)% [1]%
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Exercise XCII, problem 5, p. 385
If $M = N^{4} + x$, and $x$ is small compared with $N$, then a good approximation for $\sqrt[4]{M}$ is 5156 N + 556 MN^3 + 27Nx14(7M + 5N^4). Show that when $N = 10$, $x = 1$, this approximation is accurate to $16$ places of decimals. % [0]% (*Math. Trip.* 1886.)% [1]%
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Exercise XCII, problem 6, p. 385
Show how to sum the series _0^ P_r(n) mn x^n, where $P_{r}(n)$ is a polynomial of degree $r$ in $n$. [Express $P_{r}(n)$ in the form $A_{0} + A_{1}n + A_{2}n(n - 1) + \dots$ as in % [examples:xc]Ex. xc%. 7.]
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Exercise XCII, problem 7a, p. 385
Sum the series $\sum\limits_{0}^{\infty} n \dbinom{m}{n} x^{n}$, $\sum\limits_{0}^{\infty} n^{2} \dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.
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Exercise XCII, problem 7b, p. 385
Sum the series $\sum\limits_{0}^{\infty} n \dbinom{m}{n} x^{n}$, $\sum\limits_{0}^{\infty} n^{2} \dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.
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Exercise XCII, problem 7c, p. 385
Sum the series $\sum\limits_{0}^{\infty} n \dbinom{m}{n} x^{n}$, $\sum\limits_{0}^{\infty} n^{2} \dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.
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Exercise Misc-IX

Exercise Misc-IX, problem 1, p. 387
Given that $\log_{10} e = .4343$ and that $2^{10}$ and $3^{21}$ are nearly equal to powers of $10$, calculate $\log_{10}2$ and $\log_{10}3$ to four places of decimals. % [0]% (*Math. Trip.* 1905.)% [1]%
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Exercise Misc-IX, problem 10, p. 387
Show that $\dfrac{1}{\log(1 + x)} - \dfrac{1}{x} \to \dfrac{1}{2}$ as $x \to 0$.
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Exercise Misc-IX, problem 11, p. 387
Show that $\dfrac{1}{\log(1 + x)} - \dfrac{1}{x}$ decreases steadily from $1$ to $0$ as $x$ increases from $-1$ towards $\infty$.
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Exercise Misc-IX, problem 12, p. 387
Show that the function $(\log \xi - \log x)/(\xi - x)$, where $\xi$ is positive, decreases steadily as $x$ increases from $0$ to $\xi$, and find its limit as $x \to \xi$.
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Exercise Misc-IX, problem 13, p. 387
Show that $e^{x} > Mx^{N}$, where $M$ and $N$ are large positive numbers, f $x$ is greater than the greater of $2\log M$ and $16N^{2}$.
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Exercise Misc-IX, problem 14, p. 387
If $f(x)$ and $\phi(x)$ tend to infinity as $x \to \infty$, and $f'(x)/\phi'(x) \to \infty$, then $f(x)/\phi(x) \to \infty$. [Use the result of Ch.VI, [misc:VI]Misc. Ex. 33.] By taking $f(x) = x^{\alpha}$, $\phi(x) = \log x$, prove that $(\log x)/x^{\alpha} \to 0$ for all positive values of $\alpha$.
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Exercise Misc-IX, problem 15, p. 387
If $p$ and $q$ are positive integers then 1pn + 1 + 1pn + 2 + …+ 1qn (qp) as $n \to \infty$.
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Exercise Misc-IX, problem 16, p. 387
Prove that if $x$ is positive then $n\log\{\frac{1}{2}(1 + x^{1/n})\} \to -\frac{1}{2}\log x$ as $n \to \infty$.
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Exercise Misc-IX, problem 17, p. 387
Prove that if $a$ and $b$ are positive then 12(a^1/n + b^1/n)^n ab.
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Exercise Misc-IX, problem 18, p. 387
Show that 1 + 13 + 15 + …+ 12n - 1 = 12n + 2 + 12 + _n, where $\gamma$ is Euler’s constant (% [examples:lxxxix]Ex. lxxxix%. 1) and $\epsilon_{n} \to 0$ as $n \to \infty$.
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Exercise Misc-IX, problem 19, p. 387
Show that 1 + 13 - 12 + 15 + 17 - 14 + 19 + … = 32 2, the series being formed from the series $1 - \frac{1}{2} + \frac{1}{3} - \dots$ by taking alternately two positive terms and then one negative.
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Exercise Misc-IX, problem 2, p. 387
Determine which of $(\frac{1}{2}e)^{\sqrt{3}}$ and $(\sqrt{2})^{\frac{1}{2}\pi}$ is the greater.
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Exercise Misc-IX, problem 20, p. 387
Show that $1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \dots = \frac{1}{2}\log 2$.
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Exercise Misc-IX, problem 21, p. 387
Prove that _1^n 1(36^2 - 1) = -3 + 3_3n+1 - _n - S_n where $S_{n} = 1 + \dfrac{1}{2} + \dots + \dfrac{1}{n}$, $\Sigma_{n} = 1 + \dfrac{1}{3} + \dots + \dfrac{1}{2n - 1}$. Hence prove that the sum of the series when continued to infinity is -3 + 323 + 22. % [0]% (*Math. Trip.* 1905.)% [1]%
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Exercise Misc-IX, problem 22, p. 387
Show that _1^ 1n(4n^2 - 1) = 22 - 1, 0pt minus 3pt_1^ 1n(9n^2 - 1) = 32(3 - 1).
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Exercise Misc-IX, problem 23, p. 387
Prove that the sums of the four series _1^ 14n^2 - 1,0pt minus 3pt_1^ (-1)^n-14n^2 - 1,0pt minus 3pt_1^ 1(2n + 1)^2 - 1,0pt minus 3pt_1^ (-1)^n-1(2n + 1)^2 - 1 are $\frac{1}{2}$, $\frac{1}{4}\pi - \frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{2}\log 2 - \frac{1}{4}$ respectively.
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Exercise Misc-IX, problem 24, p. 387
Prove that $n!\, (a/n)^{n}$ tends to $0$ or to $\infty$ according as $a < e$ or $a > e$.
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Exercise Misc-IX, problem 25, p. 387
Find the limit as $x \to \infty$ of (a_0 + a_1 x + …+ a_r x^r b_0 + b_1 x + …+ b_r x^r)^_0+_1x, distinguishing the different cases which may arise. % [0]% (*Math. Trip.* 1886.)% [1]%
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Exercise Misc-IX, problem 26, p. 387
Prove that (1 + xn)0pt minus 3pt(x > 0) diverges to $\infty$. [Compare with $\sum (x/n)$.] Deduce that if $x$ is positive then (1 + x)(2 + x) …(n + x)/n! as $n \to \infty$. [The logarithm of the function is $\sum\limits_{1}^{n} \log \left(1 + \dfrac{x}{\nu}\right)$.]
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Exercise Misc-IX, problem 27, p. 387
Prove that if $x > -1$ then multline* 1(x + 1)^2 = 1(x + 1) (x + 2) + 1!(x + 1) (x + 2) (x + 3) + 2!(x + 1) (x + 2) (x + 3) (x + 4) + …. multline* % [0]% (*Math. Trip.* 1908.)% [1]%
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Exercise Misc-IX, problem 28, p. 387
No equation of the type Ae^x + Be^x + …= 0, where $A$, $B$, … are polynomials and $\alpha$, $\beta$, … different real numbers, can hold for all values of $x$.
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Exercise Misc-IX, problem 29, p. 387
Show that the sequence a_1 = e,0pt minus 3pta_2 = e^e^2,0pt minus 3pta_3 = e^e^e^3, … tends to infinity more rapidly than any member of the exponential scale.
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Exercise Misc-IX, problem 3, p. 387
Show that $\log_{10}n$ cannot be a rational number if $n$ is any positive integer not a power of $10$.
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Exercise Misc-IX, problem 30, p. 387
Prove that ddx (x)^(x) = ddx (x)^ + ddx ^(x) where $\alpha$ is to be put equal to $\psi(x)$ and $\beta$ to $\phi(x)$ after differentiation. Establish a similar rule for the differentiation of $\phi(x)^{[\{\psi(x)\}^{\chi(x)}]}$.
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Exercise Misc-IX, problem 31, p. 387
Prove that if $D_{x}^{n} e^{-x^{2}} = e^{-x^{2}} \phi_{n}(x)$ then (i) $\phi_{n}(x)$ is a polynomial of degree $n$, (ii) $\phi_{n+1} = -2x\phi_{n} + \phi_{n}'$, and (iii) all the roots of $\phi_{n} = 0$ are real and distinct, and separated by those of $\phi_{n-1} = 0$. [To prove (iii) assume the truth % [** TN: Typo in original; fixed while swapping roles of n and ] of the result for $\DPtypo{n}{\kappa} = 1$, $2$, … $\DPtypo{\kappa}{n}$, and consider the signs of $\DPtypo{\phi_{\kappa+1}}{\phi_{n+1}}$ for the $n$ values of $x$ for which $\DPtypo{\phi_{\kappa}}{\phi_{n}} = 0$ and for large (positive or negative) values of $x$.]
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Exercise Misc-IX, problem 32, p. 387
The general solution of $f(xy) = f(x)f(y)$, where $f$ is a differentiable function, is $x^{a}$, where $a$ is a constant: and that of f(x + y) + f(x - y) = 2f(x)f(y) is $\cosh ax$ or $\cos ax$, according as $f''(0)$ is positive or negative.
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Exercise Misc-IX, problem 33, p. 387
How do the functions $x^{\sin(1/x)}$, $x^{\sin^{2}(1/x)}$, $x^{\cosec(1/x)}$ behave as $x \to +0$?
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Exercise Misc-IX, problem 34, p. 387
Trace the curves $y = \tan x e^{\tan x}$, $y = \sin x \log \tan \frac{1}{2}x$.
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Exercise Misc-IX, problem 35, p. 387
The equation $e^{x} = ax + b$ has one real root if $a < 0$ or $a = 0$, $b > 0$. If $a > 0$ then it has two real roots or none, according as $a\log a > b - a$ or $a\log a < b - a$.
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Exercise Misc-IX, problem 36, p. 387
Show by graphical considerations that the equation $e^{x} = ax^{2} + 2bx + c$ has one, two, or three real roots if $a > 0$, none, one, or two if $a < 0$; and show how to distinguish between the different cases.
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Exercise Misc-IX, problem 37, p. 387
Trace the curve $y = \dfrac{1}{x} \log\left(\dfrac{e^{x} - 1}{x}\right)$, showing that the point $(0, \frac{1}{2})$ is a centre of symmetry, and that as $x$ increases through all real values, $y$ steadily increases from $0$ to $1$. Deduce that the equation 1x (e^x - 1x) = has no real root unless $0 < \alpha < 1$, and then one, whose sign is the same as that of $\alpha - \frac{1}{2}$.
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Exercise Misc-IX, problem 38, p. 387
Trace the curve $y = e^{1/x} \sqrtp{x^{2} + 2x}$, and show that the equation e^1/x x^2 + 2x = has no real roots if $\alpha$ is negative, one negative root if 0 < < a = e^1/2 2 + 22, and two positive roots and one negative if $\alpha > a$.
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Exercise Misc-IX, problem 39, p. 387
Show that the equation $f_{n}(x) = 1 + x + \dfrac{x^{2}}{2!} + \dots + \dfrac{x^{n}}{n!} = 0$ has one real root if $n$ is odd and none if $n$ is even.
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Exercise Misc-IX, problem 4, p. 387
For what values of $x$ are the functions $\log x$, $\log\log x$, $\log\log\log x$, … (*a*) equal to $0$ (*b*) equal to $1$ (*c*) not defined? Consider also the same question for the functions $lx$, $llx$, $lllx$, …, where $lx = \log |x|$.
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Exercise Misc-IX, problem 40, p. 387
Prove that if $a$ and $b$ are positive and nearly equal then ab = 12(a - b) (1a + 1b), approximately, the error being about $\frac{1}{6}\{(a - b)/a\}^{3}$.
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Exercise Misc-IX, problem 41a, p. 387
Prove by multiplication of series that if $-1 < x < 1$ then align* 12(1 + x)^2 &= 12 x^2 - 13(1 + 12)x^3 + 14(1 + 12 + 13)x^4 - …, 12(x)^2 &= 12 x^2 - 14(1 + 13)x^4 + 16(1 + 13 + 15)x^6 - …. align*
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Exercise Misc-IX, problem 42, p. 387
Prove that (1 + x)^1/x = e^1 - 12 a^2x + 124(8 + 3a)a^3x^2(1 + _x), where $\epsilon_{x} \to 0$ with $x$.
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Exercise Misc-IX, problem 43, p. 387
The first $n + 2$ terms in the expansion of $\log\left(1 + x + \dfrac{x^{2}}{2!} + \dots + \dfrac{x^{n}}{n!}\right)$ in powers of $x$ are x - x^n+1n! 1n + 1 - x1! (n + 2) + x^22! (n + 3) - … + (-1)^n x^nn! (2n + 1) . % [0]% (*Math. Trip.* 1899.)% [1]%
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Exercise Misc-IX, problem 44, p. 387
Show that the expansion of (-x - x^22 - …- x^nn) in powers of $x$ begins with the terms 1 - x + x^n+1n + 1 - _s=1^n x^n+s+1(n + s)(n + s + 1). % [0]% (*Math. Trip.* 1909.)% [1]%
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Exercise Misc-IX, problem 45, p. 387
Show that if $-1 < x < 1$ then align* 13x + 1·43·62^2x^2 + 1·4·73·6·93^2x^3 + … &= x(x + 3)9(1 - x)^7/3, 13x + 1·43·62^3x^2 + 1·4·73·6·93^3x^3 + … &= x(x^2 + 18x + 9)27(1 - x)^10/3. align*
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Exercise Misc-IX, problem 46, p. 387
Prove that align* _0^ dx(x + a)(x + b) &= 1a - b (ab), _0^ dx(x + a)(x + b)^2 &= 1(a - b)^2ba - b - b(ab), _0^ x dx(x + a)(x + b)^2 &= 1(a - b)^2 a(ab) - a + b, _0^ dx(x + a)(x^2 + b^2) &= 1(a^2 + b^2)b 12a - b(ab), _0^ x dx(x + a)(x^2 + b^2) &= 1a^2 + b^2 12b + a(ab), align* provided that $a$ and $b$ are positive. Deduce, and verify independently, that each of the functions a - 1 - a,0pt minus 3ptaa - a + 1,0pt minus 3pt12a - a,0pt minus 3pt12+ aa is positive for all positive values of $a$.
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Exercise Misc-IX, problem 47, p. 387
Prove that if $\alpha$, $\beta$, $\gamma$ are all positive, and $\beta^{2} > \alpha\gamma$, then _0^ dxx^2 + 2x + = 1^2 - + ^2 - ; while if $\alpha$ is positive and $\alpha\gamma > \beta^{2}$ the value of the integral is 1- ^2 - ^2, that value of the inverse tangent being chosen which lies between $0$ and $\pi$. Are there any other really different cases in which the integral is convergent?
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Exercise Misc-IX, problem 48, p. 387
Prove that if $a > -1$ then _1^ dx(x + a)x^2 - 1 = _0^ dtt + a = 2_1^duu^2 + 2au + 1; and deduce that the value of the integral is 21 - a^2 1 - a1 + a if $-1 < a < 1$, and 1a^2 - 1 a + 1 + a - 1 a + 1 - a - 1 = 2a^2 - 1 a - 1a + 1 if $a > 1$. Discuss the case in which $a = 1$.
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Exercise Misc-IX, problem 49, p. 387
Transform the integral $\ds\int_{0}^{\infty} \frac{dx}{(x + a) \sqrtp{x^{2} + 1}}$, where $a > 0$, in the same ways, showing that its value is 1a^2 + 1 a + 1 + a^2 + 1a + 1 - a^2 + 1 = 2a^2 + 1 a^2 + 1a + 1
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Exercise Misc-IX, problem 5, p. 387
Show that x - n1 (x + 1) + n2 (x + 2) - … + (-1)^n (x + n) is negative and increases steadily towards $0$ as $x$ increases from $0$ towards $\infty$.
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Exercise Misc-IX, problem 50, p. 387
Prove that _0^1 x dx = 14- 122.
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Exercise Misc-IX, problem 51, p. 387
If $0 < \alpha < 1$, $0 < \beta < 1$, then _-1^1 dx(1 - 2x + ^2)(1 - 2x + ^2) = 1 1 + 1 - .
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Exercise Misc-IX, problem 52, p. 387
Prove that if $a > b > 0$ then _-^ da+ b = a^2 - b^2
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Exercise Misc-IX, problem 53, p. 387
Prove that _0^1 x1 + x^2 dx = -_1^ x1 + x^2 dx,0pt minus 3pt_0^ x1 + x^2 dx = 0 and deduce that if $a > 0$ then _0^ xa^2 + x^2 dx = 2aa.
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Exercise Misc-IX, problem 54, p. 387
Prove that %[** TN: In-line in the original] _0^ (1 + a^2x^2) dx = a if $a > 0$.
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Exercise Misc-IX, problem 6, p. 387
Prove that (ddx)^n xx = (-1)^n n!x^n+1 (x - 1 - 12 - …- 1n). % [0]% (*Math. Trip.* 1909.)% [1]%
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Exercise Misc-IX, problem 7, p. 387
If $x > -1$ then $x^{2} > (1 + x) \{\log(1 + x)\}^{2}$. % [0]% (*Math. Trip.* 1906.)% [1]%
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Exercise Misc-IX, problem 8, p. 387
Show that $\{\log(1 + x)\}/x$ and $x/\{(1 + x)\log(1 + x)\}$ both decrease steadily as $x$ increases from $0$ towards $\infty$.
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Exercise Misc-IX, problem 9, p. 387
Show that, as $x$ increases from $-1$ towards $\infty$, the function $(1 + x)^{-1/x}$ assumes once and only once every value between $0$ and $1$. % [0]% (*Math. Trip.* 1910.)% [1]%
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Exercise LXXXIII

Exercise LXXXIII, problem 1, p. 360
It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from $\log x$. For when we differentiate the functional equation, first with respect to $x$ and then with respect to $y$, we obtain the two equations yf’(xy) = f’(x),0pt minus 3ptxf’(xy) = f’(y); and so, eliminating $f'(xy)$, $xf'(x) = yf'(y)$. But if this is true for every pair of values of $x$ and $y$, then we must have $xf'(x) = C$, or $f'(x) = C/x$, where $C$ is a constant. Hence f(x) = Cx dx + C’ = Cx + C’, and it is easy to see that $C' = 0$. Thus there is no solution fundamentally distinct from $\log x$, except the trivial solution $f(x) = 0$, obtained by taking $C = 0$.
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Exercise LXXXIII, problem 2, p. 360
Show in the same way that there is no solution of the equation f(x) + f(y) = f(x + y1 - xy) which possesses a differential coefficient and is fundamentally distinct from $\arctan x$.
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Exercise LXXXIV

Exercise LXXXIV, problem 1, p. 362
Between any two terms $f(x)$, $F(x)$ of the series we can insert a new term $\phi(x)$ such that $\phi(x)$ tends to $\infty$ more slowly than $f(x)$ and more rapidly than $F(x)$. [Thus between $\sqrt{x}$ and $\sqrt[3]{x}$ we could insert $x^{5/12}$: between $\sqrtp{\log x}$ and $\sqrtp[3]{\log x}$ we could insert $(\log x)^{5/12}$. And, generally, $\phi(x) = \sqrtb{f(x) F(x)}$ satisfies the conditions stated.]
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Exercise LXXXIV, problem 2, p. 362
Find a function which tends to $\infty$ more slowly than $\sqrt{x}$, but more rapidly than $x^{\alpha}$, where $\alpha$ is any rational number less than $1/2$. [$\sqrt{x}/(\log x)$ is such a function; or $\sqrt{x}/(\log x)^{\beta}$, where $\beta$ is any positive rational number.]
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Exercise LXXXIV, problem 3, p. 362
Find a function which tends to $\infty$ more slowly than $\sqrt{x}$, but more rapidly than $\sqrt{x}/(\log x)^{\alpha}$, where $\alpha$ is any rational number. [The function $\sqrt{x}/(\log\log x)$ is such a function. It will be gathered from these examples that *incompleteness* is an inherent characteristic of the logarithmic scale of infinity.]
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Exercise LXXXIV, problem 4, p. 362
How does the function f(x) = x^ (x)^’ (x)^”/ x^ (x)^’ (x)^” behave as $x$ tends to $\infty$? [If $\alpha \neq \beta$ then the behaviour of f(x) = x^- (x)^’-’ (x)^”-” [pg]363 is dominated by that of $x^{\alpha-\beta}$. If $\alpha = \beta$ then the power of $x$ disappears and the behaviour of $f(x)$ is dominated by that of $(\log x)^{\alpha'-\beta'}$, unless $\alpha' = \beta'$, when it is dominated by that of $(\log\log x)^{\alpha''-\beta''}$. Thus $f(x) \to \infty$ if $\alpha > \beta$, or $\alpha = \beta$, $\alpha' > \beta'$, or $\alpha = \beta$, $\alpha' = \beta'$, $\alpha'' > \beta''$, and $f(x) \to 0$ if $\alpha < \beta$, or $\alpha = \beta$, $\alpha' < \beta'$, or $\alpha = \beta$, $\alpha' = \beta'$, $\alpha'' < \beta''$.]
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Exercise LXXXIV, problem 5, p. 362
Arrange the functions $x/\sqrtp{\log x}$, $x\sqrtp{\log x}/\log\log x$, $x\log\log x/\sqrtp{\log x}$, $(x\log\log\log x)/\sqrtp{\log\log x}$ according to the rapidity with which they tend to infinity as $x \to \infty$.
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Exercise LXXXIV, problem 6, p. 362
Arrange x/(xx),0pt minus 3pt(x)/x,0pt minus 3ptxx/x^2 + 1,0pt minus 3ptx + 1/x(x)^2 according to the rapidity with which they tend to zero as $x \to \infty$.
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Exercise LXXXIV, problem 7, p. 362
Arrange x(1/x),0pt minus 3ptx/(1/x),0pt minus 3ptxx(1/x),0pt minus 3pt(1 - x)(1/x) according to the rapidity with which they tend to zero as $x \to +0$.
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Exercise LXXXIV, problem 8a, p. 362
Show that D_xx = 1/(xx),0pt minus 3ptD_xx = 1/(xxx), and so on.
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- 1/(xx)
verified: the printed answer passed a computed check
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- differentiate: passes 1/(x*log(x))
Exercise LXXXIV, problem 8b, p. 362
Show that D_xx = 1/(xx),0pt minus 3ptD_xx = 1/(xxx), and so on.
Printed answer:
- 1/(xxx)
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 1/(x*log(x)*log(log(x)))
Exercise LXXXIV, problem 9a, p. 362
Show that D_x(x)^ = /x(x)^1-,0pt minus 3ptD_x(x)^ = /xx(x)^1-, and so on.
Printed answer:
- /x(x)^1-
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes alpha/(x*log(x)**(1 - alpha))
Exercise LXXXIV, problem 9b, p. 362
Show that D_x(x)^ = /x(x)^1-,0pt minus 3ptD_x(x)^ = /xx(x)^1-, and so on.
Printed answer:
- /xx(x)^1-
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes alpha/(x*log(x)*log(log(x))**(1 - alpha))

Exercise LXXXV

Exercise LXXXV, problem 1, p. 366
If $dx/dy = ax$ then $x = Ke^{ay}$, where $K$ is a constant.
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Exercise LXXXV, problem 2, p. 366
There is no solution of the equation $f(y + z) = f(y)f(z)$ fundamentally distinct from the exponential function. [We assume that $f(y)$ has a differential coefficient. Differentiating the equation with respect to $y$ and $z$ in turn, we obtain f’(y + z) = f’(y)f(z),0pt minus 3ptf’(y + z) = f(y)f’(z) and so $f'(y)/f(y) = f'(z)/f(z)$, and therefore each is constant. Thus if $x = f(y)$ then $dx/dy = ax$, where $a$ is a constant, so that $x = Ke^{ay}$ (Ex. 1).]
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Exercise LXXXV, problem 3, p. 366
Prove that $(e^{ay} - 1)/y \to a$ as $y \to 0$. [Applying the Mean Value Theorem, we obtain $e^{ay} - 1 = aye^{a\eta}$, where $0 < |\eta| < |y|$.]
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Exercise LXXXVI

Exercise LXXXVI, problem 1, p. 369
Prove, by taking $y = 1$ and $n = 6$ in the inequalities (4) of [§]208, that $2.5 < e < 2.9$.
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Exercise LXXXVI, problem 2, p. 369
Prove that if $t > 1$ then $(t^{1/n} - t^{-1/n})/(t - t^{-1}) < 1/n$, and so that if $x > 1$ then _1^x dtt^1-(1/n) - _1^x dtt^1+(1/n) < 1n _1^x (t - 1t) dtt = 1n (x + 1x - 2). Hence deduce the results of [§]209.
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Exercise LXXXVI, problem 3, p. 369
If $\xi_{n}$ is a function of $n$ such that $n\xi_{n} \to l$ as $n \to \infty$, then $(1 + \xi_{n})^{n} \to e^{l}$. [Writing $n\log(1 + \xi_{n})$ in the form l (n_nl) (1 + _n)_n, and using % [examples:lxxxii]Ex. lxxxii%. 4, we see that $n\log(1 + \xi_{n})\to l$.]
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Exercise LXXXVI, problem 4, p. 369
If $n\xi_{n} \to \infty$, then $(1 + \xi_{n})^{n} \to \infty$; and if $1 + \xi_{n} > 0$ and $n\xi_{n} \to -\infty$, then (1 + _n)^n 0.
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Exercise LXXXVI, problem 5, p. 369
Deduce from (1) of [§]208 the theorem that $e^{y}$ tends to infinity more rapidly than any power of $y$.
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