In 1644, the Italian mathematician Pietro Mengoli (1625-1686) posed the question: What’s the value of the sum
First time, Leonhard Euler (1707-1783) in 1735 proved that above series converges to . In this post you can see an easy proof by using double integral that published by Tom M. Apostol in 1983 in Mathematical Intelligencer. Apostol’s Proof: Note that
and by the monotone convergence theorem we get
We change variables in this by putting , so that . Hence
where is the square with vertices and .
Exploiting the symmetry of the square we get
Now , and if then and .
It follows that and so . Hence