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