Prove that

**Solution**

Using the pentagonal number theorem we have the following:

The sum is dealt by the classic approach by summing the residues of the function

at the poles of the denominator. Hence the result.

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Prove that

**Solution**

The sum is dealt by the classic approach by summing the residues of the function

at the poles of the denominator. Hence the result.

Let us see a derivation of the series. Consider the function

as well as a circle contour of radius . Then

hence sending the contour vanishes and thus