Simple serie Q

1. Nov 29, 2004

quasar987

Hi. I am starting the study of series and I don't see how to do this problem.

"Show that

$$\sum_{n=0}^\infty \frac{1}{(n+a)(n+1+a)} = \frac{1}{a}$$"

All i got is the decomposition in partial fractions as

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

if these sum converge. I tried seeing a patern in the partial sums to find $S_n$ but it's too difficult so there must be another way.

Any hint/help will be appreciated.

Last edited: Nov 29, 2004
2. Nov 29, 2004

e(ho0n3

Using your partial fractions decomposition and rearranging the terms gives:

$$\frac{1}{a} - \frac{1}{a+1} + \frac{1}{a+1} - \frac{1}{a+2} + \ldots$$

It can't be any more obvious now can it.