# 5.5 Alternating series  (Page 2/10)

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We remark that this theorem is true more generally as long as there exists some integer $N$ such that $0\le {b}_{n+1}\le {b}_{n}$ for all $n\ge N.$

## Convergence of alternating series

For each of the following alternating series, determine whether the series converges or diverges.

1. $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{/}{n}^{2}$
2. $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}n\text{/}\left(n+1\right)$
1. Since
$\frac{1}{{\left(n+1\right)}^{2}}<\frac{1}{{n}^{2}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{1}{{n}^{2}}\to 0,$
the series converges.
2. Since $n\text{/}\left(n+1\right)↛0$ as $n\to \infty ,$ we cannot apply the alternating series test. Instead, we use the n th term test for divergence. Since
$\underset{n\to \infty }{\text{lim}}\frac{{\left(-1\right)}^{n+1}n}{n+1}\ne 0,$
the series diverges.

Determine whether the series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}n\text{/}{2}^{n}$ converges or diverges.

The series converges.

## Remainder of an alternating series

It is difficult to explicitly calculate the sum of most alternating series, so typically the sum is approximated by using a partial sum. When doing so, we are interested in the amount of error in our approximation. Consider an alternating series

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}$

satisfying the hypotheses of the alternating series test. Let $S$ denote the sum of this series and $\left\{{S}_{k}\right\}$ be the corresponding sequence of partial sums. From [link] , we see that for any integer $N\ge 1,$ the remainder ${R}_{N}$ satisfies

$|{R}_{N}|=|S-{S}_{N}|\le |{S}_{N+1}-{S}_{N}|={b}_{n+1}.$

## Remainders in alternating series

Consider an alternating series of the form

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}\text{or}\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}$

that satisfies the hypotheses of the alternating series test. Let $S$ denote the sum of the series and ${S}_{N}$ denote the $N\text{th}$ partial sum. For any integer $N\ge 1,$ the remainder ${R}_{N}=S-{S}_{N}$ satisfies

$|{R}_{N}|\le {b}_{N+1}.$

In other words, if the conditions of the alternating series test apply, then the error in approximating the infinite series by the $N\text{th}$ partial sum ${S}_{N}$ is in magnitude at most the size of the next term ${b}_{N+1}.$

## Estimating the remainder of an alternating series

Consider the alternating series

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

Use the remainder estimate to determine a bound on the error ${R}_{10}$ if we approximate the sum of the series by the partial sum ${S}_{10}.$

From the theorem stated above,

$|{R}_{10}|\le {b}_{11}=\frac{1}{{11}^{2}}\approx 0.008265.$

Find a bound for ${R}_{20}$ when approximating $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{/}n$ by ${S}_{20}.$

$0.04762$

## Absolute and conditional convergence

Consider a series $\sum _{n=1}^{\infty }{a}_{n}$ and the related series $\sum _{n=1}^{\infty }|{a}_{n}|.$ Here we discuss possibilities for the relationship between the convergence of these two series. For example, consider the alternating harmonic series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{/}n.$ The series whose terms are the absolute value of these terms is the harmonic series, since $\sum _{n=1}^{\infty }|{\left(-1\right)}^{n+1}\text{/}n|=\sum _{n=1}^{\infty }1\text{/}n.$ Since the alternating harmonic series converges, but the harmonic series diverges, we say the alternating harmonic series exhibits conditional convergence.

By comparison, consider the series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{/}{n}^{2}.$ The series whose terms are the absolute values of the terms of this series is the series $\sum _{n=1}^{\infty }1\text{/}{n}^{2}.$ Since both of these series converge, we say the series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{/}{n}^{2}$ exhibits absolute convergence.

## Definition

A series $\sum _{n=1}^{\infty }{a}_{n}$ exhibits conditional convergence    if $\sum _{n=1}^{\infty }|{a}_{n}|$ converges. A series $\sum _{n=1}^{\infty }{a}_{n}$ exhibits absolute convergence    if $\sum _{n=1}^{\infty }{a}_{n}$ converges but $\sum _{n=1}^{\infty }|{a}_{n}|$ diverges.

As shown by the alternating harmonic series, a series $\sum _{n=1}^{\infty }{a}_{n}$ may converge, but $\sum _{n=1}^{\infty }|{a}_{n}|$ may diverge. In the following theorem, however, we show that if $\sum _{n=1}^{\infty }|{a}_{n}|$ converges, then $\sum _{n=1}^{\infty }{a}_{n}$ converges.

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