Alternating Series

Introduction

With the exceptions of geometric series, where

$r$ may be negative, or the rare series with telescoping partial sums, the convergence tests we have worked with so far only work with positive-termed series. When the terms in a series can be positive or negative, things get more complicated; the sequence {

$s_n$ } of partial sums may not be monotonic, so it can be bounded yet divergent. This module will introduce the Alternating Series Test, which works on series in which the terms have alternating signs.

Alternating Series and the Alternating Series Test

An alternating series is a series $\displaystyle\sum_{n=1}^\infty a_n$ where $a_n$ has alternating signs. Notice that if $a_n$ has alternating signs, we will be able to let $b_n=\left\vert a_n \right\vert$ , and write $a_n=(-1)^n b_n$ or $a_n=(-1)^{n-1}b_n$ . For instance,

$\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac 14 + \ldots=\sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}$ has terms

$a_n=\frac{(-1)^{n-1}}{n}$ and

$b_n=\frac{1}{n}$ .

Alternating Series Test (AST): If the alternating series

$\displaystyle\sum_{n=1}^\infty (-1)^{n-1}b_n=b_1-b_2+b_3-b_4+\cdots$ satisfies

$b_n>0$ ,
$b_{n+1}\le b_n$ for all $n$ , and
$\displaystyle{\lim_{n \to \infty} b_n = 0}$ ,

then the series converges.

In other words, if the absolute values of the terms of an alternating series are non-increasing and converge to zero, the series converges.

This is easy to test; we like alternating series. To see how easy the AST is to implement, DO: Use the AST to see if

$\displaystyle\sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}$ converges. This series is called the alternating harmonic series.

This is a convergence-only test. In order to show a series diverges, you must use another test. The best idea is to first test an alternating series for divergence using the Divergence Test. If the terms do not converge to zero, you are finished. If the terms do go to zero, you are very likely to be able to show convergence with the AST.

Warning: The converse of the AST is not true; we have series that are alternating and convergent and do not satisfy the AST. For example, if we take the terms of

$\sum\frac{1}{n^2}=1+\frac{1}{2}+\tfrac14+\tfrac19+\tfrac1{16}+\tfrac1{25}+\cdots$ , and exchange the first two terms, then the second two, etc., and then put in alternating signs, we get

$\frac{1}{2}-1+\frac{1}{9}-\frac{1}{4}+\frac{1}{25}-\frac{1}{16}+\cdots$ , which does not satisfy the conditions of the AST since

$b_{n+1}\le b_n$ does not hold for all

$n$ . However, this series is convergent (we will be able to prove its convergence later using the ideas of Absolute Convergence).

The following video will explain how the AST works, give more details on the alternating harmonic series, and look at the values of some interesting alternating series.