discussing the series i want to prove the divergence of the infinite series $\sum_{n=0}^\infty \frac{(-1)^n x^n}{(n+1)^p}$  it's an alternating series so we will be dealing with the series $\sum_{n=0}^\infty \frac{A_n x^n}{(n+1)^p}$ i tries using leibnitz test but can only prove weather it converges or not 
so is it possible to compare it to the series $\sum_{n=0}^\infty \frac{1}{(n+1)^p}$  as $\frac{x^n}{(n+1)^p}>\frac{1}{(n+1)^p}$  then it diverges ?

Note that
$$\sum_{n=0}^\infty \frac{(-1)^n x^n}{(n+1)^p}$$ 
by ratio test
$$\left|\frac{(-1)^{n+1} x^{n+1}}{(n+2)^p}\frac{(n+1)^p}{(-1)^n x^n}\right|=|x|
\left(\frac{n+1}{n+2}\right)^p\to |x|$$
thus the series


*

*converges for $|x|<1$

*converges for $x=1$ by Leibniz

*for $x=-1$ by limit comparison test converges for $p>1$ and diverges otherwise

