MY MATEX NOTES

This is a short article to prove the following result To this end we will rely on some previous established results, namely: proved here, and proved here . As a corollary of our goal series we also obtain this nice series Click here  for the proof of our main series.

In this entry we will evaluate symmetric infinite sums of the type To this end we will rely on the previous established result ( proved here ) Click here for the proof.

In this entry we present a second proof via contour integration for the series (proved previously here ) Click here for the proof.

In this article we prove two remarkable results due to Ramanujan, namely and To this end we rely on the functional equation that the Dedekind´s eta function obeys, previously proved ( here ): Click here for the proof.

In this article we prove the following Ramanujan series To this end we will rely on the residues theorem.  Click here  for the proof of the Ramanujan´s series

Today we will proof the following two beautiful results found here . The first one is known to be an infinite sum proved by Ramanujan to Hardy, the second I am not sure. (1) (2) First, recall the partial fraction expansion of the hyperbolic cotangent Then we first sum becomes Since n and k are dummy variables, we can re-index the second double sum exchanging n by k and vice versa, then For the second sum