Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/239

As $n$ increases without limit, $S_{n}$ increases without limit and therefore the series is divergent.

Second Class. Oscillating series, of which

$S_{n}=1-1+1-1+\cdots +\left(-1\right)^{n-1}$

is an example. Here $S_{n}$ is zero or unity according as $n$ is even or odd, and although $S_{n}$ does not become infinite as $n$ increases without limit, it does not tend to a limit, but oscillates. It is evident that if all the terms of a series have the same sign, the series cannot oscillate.

Since the sum of a converging series is a perfectly definite number, while such a thing as the sum of a nonconvergent series does not exist, it follows at once that it is absolutely essential in any given problem involving infinite series to determine whether or not the series is convergent. This is often a problem of great difficulty, and we shall consider only the simplest cases.

136. Existence of a limit. When a series is given we cannot in general, as in the case of a geometric series, actually find the number which is the limit of $S_{n}$. But although we may not know how to compute the numerical value of that limit, it is of prime importance to know that a limit does exist, for otherwise the series may be non-convergent. When examining a series to determine whether or not it is convergent, the following theorems, which we state without proofs, are found to be of fundamental importance.^[1]

Theorem I. If $S_{n}$ is a variable that always increases as $n$ increases, but always remains less than some definite fixed number $A$, then as$n$ increases without limit, $S_{n}$ will approach a definite limit which is not greater than $A$.

Theorem II. If $S_{n}$ is a variable that always decreases as $n$ increases, but always remains greater than some definite fixed number $B$, then as $n$ increases without limit, $S_{n}$ will approach a definite limit which is not less than $B$.

Theorem III. The necessary and sufficient condition that $S_{n}$ shall approach some definite fixed number as a limit as $n$ increases without limit is that

$\lim _{n\to \infty }\left(S_{n+p}-S_{n}\right)=0$

for all values of the integer $p$.

1. See Osgood's Introduction To Infinite Series, pp. 4, 14, 64.