Here are some telescoping series problems: a. 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} ight)] b. Find the (n)th partial

Here are some telescoping series problems:

a. Verify that

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

b. 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.

c. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).