Partial fraction decomposition of the cosine

[gop]

Homework Statement

Calculate
$$\sum_{n=2}^{\infty}\frac{1}{n^{2}-1}$$
with the "standard" method and with the method of the partial fraction decomposition of the cosine.

Homework Equations

$$\pi\cot\pi z=\frac{1}{z}+\sum_{k=1}^{\infty}\frac{2z}{z^{2}-k^{2}}$$

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.

 

[HallsofIvy]

I have no idea what you mean by the "partial fraction decomposition of the cosine".

 

[gop]

Well it should actually read "partial fraction decomposition of the cotangent" (thus the formula)

The idea is to write the cotangent as a infinite series of partial fractions (because there are infinite many zero's of the denominator of cos(x)/sin(x)). If you do this you arive at the formula given above.

 

[Gib Z]

Of course it won't work for integers. When you take the k=z term of the sum, what happens?

 

[gop]

yeah I know that it doens't work for integers so I have to rewrite it somehow so that I can calculate the given series with the formula (as is stated in the exercise). The problem is I don't really know how to do that.