Answer
Convergent
Work Step by Step
Let us suppose that a series $\Sigma a_n $ and $a_n=(-1)^n b_n$ or, $a_n=(-1)^{n+1} b_n$ with $b_n \geq 0$ for all $n$. We will apply the Alternating Series Test to compute when an infinite series converges, it must follow the following two conditions:
a) $\lim\limits_{n \to \infty}b_n=0$
b) $b_n$ shows a decreasing sequence.
Here, we have the $b_n=\dfrac{n+1}{n^2+1}$
a) $\lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty} \dfrac{n+1}{n^2+1}=\lim\limits_{n \to \infty} \dfrac{1+1/n}{1+1/n^2} =0$
b) We can see that $b_{n+1}=\dfrac{n+1}{(n+1)^2+1} \leq b_n=\dfrac{n+1}{n^2+1}$. This means that $b_n$ is a decreasing sequence.
So, both the above conditions are satisfied. Thus, the given series is convergent by the Alternating Series Test.
