The given series is \(\sum_{n=1}^\infty\frac{(-1)^n}{n^5}\)

To check its convergence or divergence using Alternating series test.

Solution:

The alternating series says that if we have series of form, \(\sum_{n=1}^\infty(-1)^nb_n\), then if,

1)\(\lim_{n\rightarrow\infty}b_n=0\) and

2)\(b_n\) is a decreasing sequence, then the series \(\sum_{n=1}^\infty(-1)^nb_n\) is said to be convergent.

Since we have series \(\sum_{n=1}^\infty\frac{(-1)^n}{n^5}\) we have sequence \(b_n\) as \(b_n=\frac{1}{n^5}\),

Now to check both the condition using alternating series test for convergence,

1) \(\lim_{n\rightarrow\infty}b_n=\lim_{n\rightarrow\infty}\frac{1}{n^5}=0\)

2) \(n^5<(n+1)^5\)

\(\frac{1}{n^5}>\frac{1}{(n+1)^5}\)

\(b_n>b_{n+1}\)

So \(b_n\) is the decreasing sequence as well,

Since , both the condition are satisficed so the given series is convergent.

Hence, the given series \(\sum_{n=1}^\infty\frac{(-1)^n}{n^5}\) is convergent.