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

## Answer

\[\lim_{n \rightarrow \infty}\frac{\frac{n+2}{n^2+n+1}}{\frac{1}{n}}=1,\]

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.

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