Use Laplace transforms to sum the following series or write as a single integral.

a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{1+2 n}\).

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

c. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(n+3)}\).

d. \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n^{2}-a^{2}}\).

e. \(\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}-a^{2}}\).

f. \(\sum_{n=1}^{\infty} \frac{1}{n} e^{-a n}\).