# 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.

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

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.

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

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.