Understanding Telescoping Series: Finding the Sum

[qeteshchl]

My apologies beforehand for not using the right format for this post.

Homework Statement

Find the sum of (from 1 to inf) of \sum8/(n(n+1)(n+2))

Homework Equations

The Attempt at a Solution

I approached the problem like I would a telescoping series by using partial faction decomposition to split it up. I arrived at:

\sum4/n - 8/(n+1) + 4/(n+2)

I started plugging in numbers for 1 to try and arrive at a pattern:

(1 - 1 + 1/3) + (1/2 - 2/3 + 1/4) + (1/3 - 2/4 + 1/5) + (1/4 - 2/5 + 1/6) ...

I'm not seeing any discernible pattern to what is canceling out. Am I just approaching this problem wrong? Thanks for any help you guys may provide!

 

[LCKurtz]

My apologies beforehand for not using the right format for this post.

Homework Statement

Find the sum of (from 1 to inf) of \sum8/(n(n+1)(n+2))

Homework Equations

The Attempt at a Solution

I approached the problem like I would a telescoping series by using partial faction decomposition to split it up. I arrived at:

\sum4/n - 8/(n+1) + 4/(n+2)

I started plugging in numbers for 1 to try and arrive at a pattern:

(1 - 1 + 1/3) + (1/2 - 2/3 + 1/4) + (1/3 - 2/4 + 1/5) + (1/4 - 2/5 + 1/6) ...

I'm not seeing any discernible pattern to what is canceling out. Am I just approaching this problem wrong? Thanks for any help you guys may provide!

I'm assuming your partial fraction expansion is OK. What I notice in your general term is the two 4's and the -8. If they went with the same n, they would cancel. But the 4/n will be 4/(n+1) in the next term and the 4/(n+2) will be 4/(n+1) in the previous term. Try looking for that cancellation pattern.

 

[qeteshchl]

I don't know if it was the most efficient way of doing it but I stretched out the nth term values pretty far until I found a pattern. Thanks for your advice LCKurtz, really appreciate the response!