Originally Posted by

**Twig** Hi

Thanks for your help guys, it became a bit clearer for me.

I guess the 'trick' here is to write out some terms when $\displaystyle a_{k}$

has been partial divided, and see what terms cancel out each other?

I got:

$\displaystyle \frac{1}{2}(\frac{1}{2k - 1} - \frac{1}{2k + 1}) + \frac{1}{2}(\frac{1}{2k + 1} - \frac{1}{2k+3})\, ... $

I can see that some terms cancel each other.

So I guess I`ll end up with:

$\displaystyle \frac{1}{2}(1 - \frac{1}{2n + 1}) \rightarrow \frac{1}{2}\; \mbox{when}\; n\rightarrow \infty $

Also thanks Krizalid for helping me with the convergence proof.

One question however, why did you write

$\displaystyle \frac{1}{3} + \sum_{k=1}^{\infty}\, \frac{1}{(2k+1)(2k+3)} $ ?

Couldnīt we just make proof from scratch, without explicitly writing out the first term $\displaystyle \frac{1}{3} $ ?