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