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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 $n\ge N.$
For each of the following alternating series, determine whether the series converges or diverges.
Determine whether the series $\sum}_{n=1}^{\infty}{\left(\mathrm{-1}\right)}^{n+1}n\text{/}{2}^{n$ converges or diverges.
The series converges.
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
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
Consider an alternating series of the form
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
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 ${b}_{N+1}.$
Consider the alternating series
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 $\sum}_{n=1}^{\infty}{\left(\mathrm{-1}\right)}^{n+1}\text{/}n$ by ${S}_{20}.$
$0.04762$
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 $\sum _{n=1}^{\infty}{(\mathrm{-1})}^{n+1}}\text{/}n.$ The series whose terms are the absolute value of these terms is the harmonic series, since $\sum _{n=1}^{\infty}|}{(\mathrm{-1})}^{n+1}\text{/}n|={\displaystyle \sum _{n=1}^{\infty}1}\text{/}n.$ 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 $\sum _{n=1}^{\infty}{(\mathrm{-1})}^{n+1}}\text{/}{n}^{2}.$ 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 $\sum _{n=1}^{\infty}{(\mathrm{-1})}^{n+1}}\text{/}{n}^{2$ exhibits absolute convergence.
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 $\sum}_{n=1}^{\infty}\left|{a}_{n}\right|$ 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 $\sum}_{n=1}^{\infty}\left|{a}_{n}\right|$ converges, then $\sum}_{n=1}^{\infty}{a}_{n$ converges.
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