How to deal with series of the form $\sum^{\infty}_{n=1}f(n)^{(-1)^n}$

Question

I have no idea on how to prove series like $$\sum^{\infty}_{n=1}\left(\frac{1}{n^x}\right)^{(-1)^n}$$ for natural $x$ -divergent or convergent. By intuition it seems they diverge .

Answer 1

Marking $k=n/2$ the sum can be written as (without $n=1$, which is not relevant to convergence): $$\sum_{k=1}^\infty\frac{1}{(2k)^x}+{(2k+1)}^x$$ Which is clearly diverges.

Answer 2

Hint

A necessary condition for the convergence of the series $\displaystyle \sum_n a_n$ is that $\displaystyle \lim_{n\to\infty} a_n=0$.