5.5 Alternating series  (Page 17/10)

So far in this chapter, we have primarily discussed series with positive terms. In this section we introduce alternating series—those series whose terms alternate in sign. We will show in a later chapter that these series often arise when studying power series. After defining alternating series, we introduce the alternating series test to determine whether such a series converges.

The alternating series test

A series whose terms alternate between positive and negative values is an alternating series    . For example, the series

and

are both alternating series.

Series (1), shown in [link] , is a geometric series. Since $\left|r\right|=\left|\text{−}1\text{/}2\right|<1,$ the series converges. Series (2), shown in [link] , is called the alternating harmonic series. We will show that whereas the harmonic series diverges, the alternating harmonic series converges.

To prove this, we look at the sequence of partial sums $\left\{S_k\right\}$ ( [link] ).

Proof

Consider the odd terms $S_{2k+1}$ for $k\ge 0.$ Since $1\text{/}(2k+1)<1\text{/}2k,$

Therefore, $\left\{S_{2k+1}\right\}$ is a decreasing sequence. Also,

Therefore, $\left\{S_{2k+1}\right\}$ is bounded below. Since $\left\{S_{2k+1}\right\}$ is a decreasing sequence that is bounded below, by the Monotone Convergence Theorem, $\left\{S_{2k+1}\right\}$ converges. Similarly, the even terms $\left\{S_{2k}\right\}$ form an increasing sequence that is bounded above because

and

Therefore, by the Monotone Convergence Theorem, the sequence $\left\{S_{2k}\right\}$ also converges. Since

we know that

Letting $S=\underset{k\to \infty }{\text{lim}}{S}_{2k+1}$ and using the fact that $1\text{/}\left(2k+1\right)\to 0,$ we conclude that $\underset{k\to \infty }{\text{lim}}{S}_{2k}=S.$ Since the odd terms and the even terms in the sequence of partial sums converge to the same limit $S,$ it can be shown that the sequence of partial sums converges to $S,$ and therefore the alternating harmonic series converges to $S.$

It can also be shown that $S=\text{ln}2,$ and we can write

□

More generally, any alternating series of form (3) ( [link] ) or (4) ( [link] ) converges as long as $b_1\ge b_2\ge b_3\ge \text{⋯}$ and $b_n\to 0$ ( [link] ). The proof is similar to the proof for the alternating harmonic series.