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Determine whether the series $$\sum\limits_{n=1}^\infty \frac{n!+n}{(n+2)!}$$ is convergent or divergent.

Wolfram Alpha says that "By the comparison test, the series converges" but I can't find any good possibilities for a bounding series.

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  • $\begingroup$ Try the series with terms $2n!/(n+2)!$. $\endgroup$ – David Mitra Dec 7 '14 at 15:51
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$$\sum_n \frac{n!+n}{(n+2)!} = \sum_n \frac{n!}{(n+2)!} + \sum_n \frac{n}{(n+2)!} \leq \sum_n \frac{1}{(n+1)(n+2)} + \sum_n \frac{n+2}{(n+2)!}$$ and the two series converge.

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Compare the sum, after say, $n = 20$, with $\sum \frac{20}{n^2}$. Better still, note that it's less than $$ \sum \frac{n! + n!}{(n+2)!} $$ and compare it to $\sum \frac{2}{n^2}$.

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