Consider the sum \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}\).

a. Use an appropriate convergence test to show that this series converges.

b. Verify that

\[\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}=\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right)\]

c. Find the \(n\)th partial sum of the series \(\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right)\) and use it to determine the sum of the resulting telescoping series.