How Can You Use Partial Fractions to Find the Sum of an Infinite Series?

[Sean1218]

Homework Statement

Ʃ 4/(n(n+2)) from n=1 to n=infinity

Homework Equations

The Attempt at a Solution

I tried using partial fractions to get A/n + B/(n+2), and I solved for A and B to get A=2 and B=-2

I tried summing them up, so everything would cancel except the first & last term, but nothing cancels.

I'm not sure of any other methods for finding sums or if I'm not just using this one wrong.

Any help?

 

[HallsofIvy]

Nothing cancels? Yes, partial fractions gives the addend as 2/n- 2/(n+2).
When n= 1, that is 2- 2/3.
When n= 2, that is 1- 2/4.
When n= 3, that is 2/3- 2/5.
When n= 4, that is 2/4- 2/6.
When n= 5, that is 2/5- 2/7.
When n= 6, that is 2/6- 2/8.
When n= 7, that is 2/7- 2/9
etc.

I see a lot cancelling!

 

[LCKurtz]

You have the right idea. Try writing out the first 6 or so terms of$$
\sum_1^\infty(\frac 2 n - \frac 2 {n+2})$$leaving in the parentheses and not simplifying as you go. You will see some terms that cancel in the middle. Once you do that write the first $n$ terms.

[Edit]I see Halls types faster than I do.