Use Laplace transforms to prove

\[\sum_{n=1}^{\infty} \frac{1}{(n+a)(n+b)}=\frac{1}{b-a} \int_{0}^{1} \frac{u^{a}-u^{b}}{1-u} d u\]

Use this result to evaluate the following sums:

a. \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\).

b. \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+3)}\).