Two Beautiful Mathematical Results - Part 2

In Part 1 of this article (see here) we featured the two results below, as well as a simple way to prove these formulas.

Here, we continue on the same topic, featuring and proving the formulas below, which are just the tip of the iceberg.

However cool these formulas might look, the biggest contribution here is a general framework to solve much more general problems of this type. The mathematical level is still relatively simple, accessible to people in their first year of college education, if they attended a solid course on calculus. These results are still easy to prove for people who have been exposed to the basics of complex number theory.

I haven't seen these results published anywhere, but my guess is that they are not new. I encourage readers to post questions on Quora or Stackexchange to find more references on this topic, as Google search is of no use here.The focus here is to get more people interested in mathematics, by featuring fascinating results that are not that hard to prove, even for high school students participating in mathematical Olympiads. Also, it could be an interesting, fresh, original topic for university professors to discuss in their lectures or for exams. Finally, if you have seen many similar formulas on Wikipedia (or elsewhere), and you are wondering how they are derived, you will find the solution in this article.

The last section introduces a new, tough, unrelated problem, still unsolved today, that will be of interest to people with a background in probability and/or number theory.

Read the full article.