(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

s=[itex]\sum^{k=\infty}_{k=1}[/itex] [itex]\frac{2}{k^{2}+10k+24}[/itex]

2. Relevant equations

3. The attempt at a solution

I can pull the 2 out. I can set n=[itex]\infty[/itex] and just take the limit as n approaches infinity of the sum from 1 to n.

I use partial fractions to write out:

[itex]\frac{1}{(n+4)(n+6)}[/itex]=[itex]\frac{1}{2n+8}[/itex]-[itex]\frac{1}{2n+12}[/itex]

To see it better, I write out the first few terms manually. I get the following (as n approaches infinity of course):

[itex]\frac{s}{1}[/itex]=[itex]\frac{1}{5}[/itex]-[itex]\frac{1}{7}[/itex]+[itex]\frac{1}{6}[/itex]-[itex]\frac{1}{8}[/itex]+[itex]\frac{1}{7}[/itex]-[itex]\frac{1}{9}[/itex]+[itex]\frac{1}{8}[/itex]-[itex]\frac{1}{10}[/itex]+...+[itex]\frac{2}{2n+8}[/itex]-[itex]\frac{2}{2n+12}[/itex]

But I am very confused. It is similar to a telescoping series.. but I am not used to something like this. Usually the 2nd term cancels out the third, the 4rth with the 5th, and so on and so forth until all you have left is the first and the very last term. But in THIS series, the 2nd cancels our the 5th, the 4th with the 7th, etc. And I can't seem to work out what would remain.. Can someone please help me?

BTW I know the answer to be 11/30, in case that helps at all.

Thanks in advance!

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# Homework Help: Please help me solve this simple infinite sum

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