Alternating Series

Another type of series is an alternating series. This is a series where the sign of each term in the sequence switches at each step. For example,

$$\displaystyle \sum^{\infty}_{n=1}(-1)^n\displaystyle \frac{1}{n}\;\;$$and$$\;\; \displaystyle \sum^{\infty}_{n=1}\displaystyle \frac{\sin[(2n+1)\pi/2]}{2n!}$$

are both alternating series.

An alternating series converges if and only if the following two conditions hold.

1. $$\lim_{n\to\infty}a_n=0$$;

2. $a_{n+1}\le a_n$ for all $n$.

So the alternating series $\sum^{\infty}_{n=1}(-1)^n(1/n)$ converges and the alternating series $\sum^{\infty}_{n=1}(-1)^n(n+1)/(3n-7)$ diverges. It is important to be careful here. If $\lim_{n\to\infty}a_n=0$ for an alternating series, then the series will converge. You can not use this to test for convergence of a non-alternating series.

If the positive counterpart of an alternating series converges then we say the series converges absolutely. Otherwise the alternating series converges conditionally. For example,

$$\displaystyle \sum^{\infty}_{n=1}(-1)^n\displaystyle \frac{1}{n}$$

converges conditionally and the series

$$\displaystyle \sum^{\infty}_{n=1}(-1)^n\left(\displaystyle \frac{1}{2}\right)^n$$

converges absolutely.

We can always determine the value of an infinite alternating series within a certain degree of accuracy. If you have added together $n$ terms, then the error is no more than the $n+1$ term in the sequence; i.e., $\vert S-S_n\vert\le a_{n+1}$ where $S$ is the value of the series. For example, take the series $$\sum^{\infty}_{n=1}(-1)^n1/n$$. Then $S_5=1-1/2+1/3-1/4+1/5=47/60.$ This within $a_6=1/6$ of the actual sum.

$$37/60=47/60-1/6\le\displaystyle \sum^{\infty}_{n=1}(-1)^n\displaystyle \frac{1}{n} \le 47/60+1/6=57/60$$

If we need to be more accurate we simply need to go out further in the series.