## An series related to the eta Dedekind function

Prove that

**Solution**

Let us consider the function

and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,

We note that and that .

It’s also easy to see that

Hence, as we have that

By Residue theorem we have that

It is straightforward to show that

Hence,

in the limit . The result follows.

## Fourier transformation

Let and . Show that

**Solution**

We note that

Thus,

## Value of a numeric expression

In a triangle it holds that

(1)

Prove that

## Triangle relation

Prove that in any triangle it holds that

## Convergence of series

Let denote the greatest prime factor of . For example , . Define . Examine if the sum

converges.