Conclude if the given series converges or diverges and justify the conclusion

by RoRi

February 29, 2016

Test the following series for convergence or divergence. Justify the decision.

$\sum_{n=1}^{\infty} \frac{n!}{(n+2)!}.$

This series converges. First,

$\sum_{n=1}^{\infty} \frac{n!}{(n+2)!} = \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)}.$

Then, using the comparison test

$\frac{1}{(n+1)(n+2)} < \frac{1}{n^2} \qquad \text{for all } n$

implies the series converges since $\sum \frac{1}{n^2}$ converges.

Stumbling Robot