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

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

Answer

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. 

The answer is accepted.