Absolute and conditional convergence of series with parameter

Question

I have the following series:

$$\sum_{n=2}^{\infty}\frac{(-1)^n}{(n+(-1)^n)^p}$$

I need to check for absolute and conditional convergence values, depending on the parameter $p$. Any tips on how to solve this problem?

Answer 1

Hint: $$\frac{1}{(n+1)^p}<\left|\frac{(-1)^n}{(n+(-1)^n)^p}\right|<\frac{1}{(n-1)^p}.$$If you can establish these inequalities, then the rest (for absolute convergence) can be determined via the integral test for series. To check the series for conditional convergence, use the ideas above to determine what $p$ is allowed to be such that you'll still have $$\frac{1}{(n+(-1)^n)^p}\to0\quad\text{as}\quad n\to\infty.$$