Series

Excerpts

• Series

  The general mathematical idea of a series is that of a set of things ranged in order, that is, in sequence; This meaning is accurately represented in the common use of the term.

• Series

  It is evident that nothing that has been said gives the slightest idea as to how the “sum to infinity” of a series is to be found.

• Series

  But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates; and in this sense there is no such thing as the sum of an infinite series.

• Series

  When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations.

• Series

  This decimal is merely a way of symbolizing the “sum to infinity” of the series

• Series

  The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach.

• Series

  But this description of the meaning of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics.

• Series

  Mathematics would be a much easier science than it is, if this were the case. Unfortunately the supposition is not true.

• Series

  The statesman in framing his speech puts the dominating issues first and lets the details fall naturally into their subordinate places.

• Series

  It is easy to verify in the case of small values of $n$ that $n!$ is the number of ways of arranging $n$ things in order.

• Series

  The importance of the exponential function is that it represents any changing physical quantity whose rate of increase at any instant is a uniform percentage of its value at that instant.

• Series

  The curve, which is something like a cocked hat, is called the curve of normal error.

Equations

• Series

  n × (n - 1) × (n - 2) × (n - 3) × \dots × 4 × 3 × 2 × 1\Add{,}

  The number of permutations of n things is the product of the first n integers, which is written n!.

• Series

  s_{n} = u_{1} + u_{2} + u_{3} + \dots + u_{n}

  The sum s_n of the first n terms of a series is the sum of the terms u_1 through u_n.

• Series

  \tfrac{1}{9} = .1 + \tfrac{1}{90}

  The recurring decimal .1111... equals 1/9, shown by writing 1/9 as the first term .1 plus the remainder 1/90.

• Series

  s_{n} = 1 + x + x^{2} + x^{3} + \dots + x^{n}

  The sum of the first n terms of the geometrical series in x is s_n.

• Series

  s_{n} = \frac{1 - x^{n+1}}{1 - x}

  For x not equal to 1, the sum of n terms of the geometric series equals (1 - x^(n+1))/(1 - x).

• Series

  \frac{1}{1 - x} = 1 + x + x^{2} + \dots + x^{n} + \dots

  For |x| < 1 the geometric series 1 + x + x^2 + ... converges, with sum 1/(1 - x).

• Series

  \exp x = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{n}}{n!} + \dots

  The exponential function exp x is defined as the sum to infinity of the exponential series.

• Series

  (\exp x) × (\exp y) = \exp(x + y)

  The product of exp x and exp y equals exp(x + y); this is the addition-theorem of the exponential function.

• Series

  \sin(x + y) = \sin x \cos y + \cos x \sin y

  The sine of a sum equals sin x cos y plus cos x sin y.

• Series

  \cos(x + y) = \cos x \cos y - \sin x \sin y

  The cosine of a sum equals cos x cos y minus sin x sin y.

• Series

  \sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \text{etc.} \dots

  The sine is defined as the limit of the alternating power series in x.

• Series

  \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \text{etc.} \dots

  The cosine is defined as the limit of the alternating power series in x.

• Series

  y = \exp(-x^{2})

  The curve of normal error is the graph of exp(-x^2), whose function is central to statistics.

• Series

  y = \exp(-cx) × \sin \frac{2\pi x}{p}

  A damped vibration: a sine of period p multiplied by an exponential decay exp(-cx) with constant percentage damping.

Problems

No exercises in this chapter.