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 
is divergent. Also since 
by the Alternating Series Test

converges conditionally but not absolutely. 

The answer is accepted.
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