Question:
Hello, I'm currently studying series in my calculus course and I've come across a problem that I'm having trouble with. The problem is to determine whether the following series is absolutely convergent or not:
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n+x^2}$$
where $-\infty<x<\infty$.
My Attempt:
I know that a series is absolutely convergent if the series of absolute values of its terms is convergent. So, I tried to apply the definition of absolute convergence to this series:
$$\sum_{n=1}^{\infty}\left|\frac{(-1)^n}{n+x^2}\right| = \sum_{n=1}^{\infty}\frac{1}{n+x^2}$$
However, I'm not sure how to proceed from here. I thought about using the comparison test, but I couldn't find a suitable series to compare it with.
Background:
I'm an undergraduate student taking a course in calculus. I'm familiar with the definitions of series convergence, absolute convergence, and the comparison test.
Context:
This problem is from my textbook (unfortunately, I don't have the exact reference at the moment), and it's in the section on series convergence. I'm trying to understand how to determine absolute convergence for different types of series.
