Find The Sum Of The Convergent Series

[Bashyboy]

Homework Statement

\sum_{n=2}^{\infty} \frac{1}{n^2-1}

Homework Equations

The Attempt at a Solution

After doing partial fraction decomposition, I discovered that it was a telescoping series of some sort; the partial sum being 1/2[ (1 -1/3) + (1/2 - 1/4) + (1/3 - 1/4) +...] The only thing I can't do is see the pattern to make a nth partial sum. How would I go about this?

 

[vela]

Write out the first few partial sums explicitly. Four or five is probably enough to see how the cancellations work out and which terms are going to stick around.

 

[Bashyboy]

Well, I can see that the 1 and 1/2 would have nothing to cancel out with; but what I can truly see doesn't go beyond that.

 

[vela]

What specifically is confusing you? Frankly, I don't see how you can not see the pattern after writing out the first handful of partial sums.

 

[Bashyboy]

I attached the solution from the text-book. I don't understand how they get 1/n in the simplified nth partial sum formula.