INFINITE SUM FROM GRADSHTEYN AND RYZHIK I
I saw the following infinite sum in the monumental Table of Integrals, Series, and Products of Gradshteyn and Ryzhik in entry 0.238 3. and wanted to proof it. Since it wasn´t that bad, I thought of a variation, namely This one was way harder than the first, but unfortunately it does not appear in GR to check if my computation matched with the correct answer. Then, I went to the infinite sum calculator of Wolfram , and was lucky enough that for this sum it returned back a closed form that matched with the calculations! So lets get to it! We will rely on the Digamma function to evaluate both sums, so it´s good to recall the following properties : (1) (2) (3) For the first sum the biggest challenge is the partial fraction decomposition. Once We get through it, the Digamma function technique take´s care. From equation (1),(2) and (3) above and this post we get And finally! (4) ...