Basel problem

I was thinking about the sum of the infinite series,

Sigma(1/N^2) which sums to PI^2 /6.

I heard this mentioned in the middle of a lecture and tried to solve the problem. When I didn't get any good idea, I cheated and looked at wikipedia to realise it is called Basel Problem.

It has an amazing history behind it. This is the first problem which made Euler famous.
I also his proof given in the wiki page, is elegant [but not rigorous my modern standards]. But more importantly I was amazed to know its value eluded the best of mathematicians of his day! and here was me, sitting and trying to derive the result in 10 minutes.

By the way, generalisation of this is called Riemann Zeta function, with this having value Z(2). So I guess you can think of the generalisation yourself.