# Convergence of $\sum\limits_{n=1}^{\infty}(-1)^n\frac{n+2}{n^2+n+1}$

Determine if the following series converge absolutely, conditionally, or diverge

$\sum\limits_{n=1}^{\infty}(-1)^n\frac{n+2}{n^2+n+1}.$

First note that
$\lim_{n \rightarrow \infty}\frac{\frac{n+2}{n^2+n+1}}{\frac{1}{n}}=1,$
and hence by the Limit Comparision Test
$\sum_{n=1}^{\infty}\frac{n+2}{n^2+n+1}$
is divergent. Also since
$\lim_{n\rightarrow}\frac{n+2}{n^2+n+1}=0,$
by the Alternating Series Test

$\sum_{n=1}^{\infty}(-1)^n\frac{n+2}{n^2+n+1}$
converges conditionally but not absolutely.