Chapter 11 - Infinite Sequences and Series - Review - Exercises - Page 803: 35

$0.9721$

$\Sigma_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{5}}=1-\frac{1}{32}+\frac{1}{243}-\frac{1}{1024}+...$ Since this is an alternating series we can use the alternating series estimation theorem, Here, $a_n=\frac{(-1)^{n+1}}{n^{5}}$ which means we only need to add up $a_{1}$ to $a_{6}$ $s_{6}=\Sigma_{n=1}^{6}\frac{(-1)^{n+1}}{n^{5}}\approx 0.9721$