## A logarithmic integral

Let . Prove that:

**Solution**

Using the known Fourier series as well as the known generating function of the Catalan numbers

Then, we have successively:

## A Gamma summation

Let and . Prove that

**Solution**

Consider the function

We easily evaluate that

Hence it follows from Poisson summation formula that

Thus,

The result follows.

## A Gudermannian integral

Let . Prove that

where is the Gudermannian function.

**Solution**

First of all we note that and . Hence,

## An odd trigonometric integral

Let . Prove that

**Solution**

Since we deduce that

Hence,

since and . Finally, we note that the sum telescopes hence the result.

## A rational harmonic series

Can the rational numbers in the interval be enumerated as a sequence in such a way that the series is convergent?