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## Homework Statement

Calculate

[tex]\sum_{n=2}^{\infty}\frac{1}{n^{2}-1}[/tex]

with the "standard" method and with the method of the partial fraction decomposition of the cosine.

## Homework Equations

[tex]\pi\cot\pi z=\frac{1}{z}+\sum_{k=1}^{\infty}\frac{2z}{z^{2}-k^{2}}[/tex]

## The Attempt at a Solution

The "standard" method wasn't a problem just partial fraction decomposition and some index-shifting. the result is 3/4.

However, when it comes to the cosine function my problem is that I can't plug integers (i.e. 1) in the function because it isn't defined there. Thus, i tried rewriting the term; however, the only rewriting of the term that leads to a result is to do the usual partial fraction decomposition which sort of defeats the purpose.