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# Tag Archives: Series

## Peculiar alternating series

Prove that

**Solution**

We simply note that

## Hyperbolic series

Evaluate the series

**Solution**

We have successively

since by the unique factorization theorem, any positive even integer can be written in a unique way as with and .

## 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.

## Factorial series

Prove that

**Solution**

We have successively: