5.5 Alternating series  (Page 2/10)

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 $n\ge N.$

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

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

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 $b_{N+1}.$

Absolute and conditional convergence

Consider a series ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ and the related series ${\displaystyle \sum _{n=1}^{\infty }|}{a}_n|.$ Here we discuss possibilities for the relationship between the convergence of these two series. For example, consider the alternating harmonic series ${\displaystyle \sum _{n=1}^{\infty }{(\mathrm{-1})}^{n+1}}\text{/}n.$ The series whose terms are the absolute value of these terms is the harmonic series, since ${\displaystyle \sum _{n=1}^{\infty }|}{(\mathrm{-1})}^{n+1}\text{/}n|={\displaystyle \sum _{n=1}^{\infty }1}\text{/}n.$ 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 ${\displaystyle \sum _{n=1}^{\infty }{(\mathrm{-1})}^{n+1}}\text{/}{n}^2.$ The series whose terms are the absolute values of the terms of this series is the series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^2.$ Since both of these series converge, we say the series ${\displaystyle \sum _{n=1}^{\infty }{(\mathrm{-1})}^{n+1}}\text{/}{n}^2$ exhibits absolute convergence.

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