Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/238

We have so far discussed only a particular series (A) when the number of terms increases without limit. Let us now consider the general problem, using the series

(C)	$u_{1},\,u_{2},\,u_{3},\,u_{4},\,\cdots$,

whose terms may be either positive or negative. Denoting by $S_{n}$ the sum of the first $n$ terms, we have,

$S_{n}=u_{1}+u_{2}+u_{3}+\cdots +u_{n}$,

and $S_{n}$ is a function of $n$. If we now let the number of terms ($=n$) increase without limit, one of two things may happen: either

Case I. $S_{n}$ approaches a limit, say $u$, indicated by

$\lim _{n\to \infty }S_{n}=u$; or

Case II. $S_{n}$ approaches no limit.

In either case (C) is called an infinite series. In Case I the infinite series is said to be convergent and to converge to the value $u$, or to have the value $u$, or to have the sum $u$. The infinite geometric series discussed at the beginning of this section is an example of a convergent series, and it converges to the value 2. In fact, the simplest example of a convergent series is the infinite geometric series

$a,\,ar,\,ar^{2},\,ar^{3},\,ar^{4},\,\cdots ,$

where $r$ is numerically less than unity. The sum of the first $n$ terms of this series is, by 6, § 1,

$S_{n}={\frac {a\left(1-r^{n}\right)}{1-r}}={\frac {a}{1-r}}-{\frac {ar^{n}}{1-r}}$.

If we now suppose $n$ to increase without limit, the first fraction on the right-hand ,side remains unchanged, while the second approaches zero as a limit. Hence

$\lim _{n\to \infty }S_{n}={\frac {a}{1-r}}$,

a perfectly definite number in any given case.

In Case II the infinite series is said to be nonconvergent^[1]. Series under this head may be divided into two classes.

First Class. Divergent series, in which the sum of $n$ terms increases indefinitely in numerical value as $n$ increases without limit; take for example the series

$S_{n}=1+2+3+\cdots +n$.

1. Some writers use divergent as equivalent to nonconvergent