discussing the series

Question

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 ?

Answer 1

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