Hey all, I'm having a bit of trouble on this one:

Suppose $\displaystyle \sum^{\infty}_{1} a_{k}$ is a series with partial sums $\displaystyle S_{n} = \frac{1}{2n +1}, n \in N$

a) Explain why $\displaystyle \sum^{\infty}_{1} a_{k}$ is convergent and find the sum of the series.

b) Find $\displaystyle a_{k}$, k = 1, 2, ...

Some of the scratch work I've been doing:

$\displaystyle S_{1} = \frac{1}{3} = a_{1}$

$\displaystyle S_{2} = \frac{1}{5} = a_{1} + a_{2}$

$\displaystyle S_{3} = \frac{1}{7} = a_{1} + a_{2} + a_{3}$ ...

$\displaystyle a_{1} = \frac{1}{3}$

$\displaystyle a_{2} = -\frac{2}{15}$

$\displaystyle a_{3} = -\frac{2}{35}$

$\displaystyle \sum^{\infty}_{1} a_{k} = \frac{1}{3} - 2(\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + ... + \frac{1}{(2n+1)(2n+3)} + ... )$

Something is obviously not jumping out at me. Any help would be appreciated.