Sum of Series?

Question

Why is the sum of k=1 to k=infinity of 1/x^2 equal to pi^2/6?

Answer 1

the quickest way I know to show this uses the theory of fourier series. If you have access to material about that subject, check under "parseval's identity".

It is possible to prove it using more elementary means, but the proof is more complicated, since you have to work harder with less sophisticated tools. the mathematician who is credited with first figuring this out was Leonhard Euler. Here is a link to how he did it:

http://www.imakenews.com/eletra/mod_print_view.cfm?this_id=900959&u=ptcexpress&show_issue_date=F&issue_id=000206628&lid=b11&uid=0&XXDESXXpower=F

Answer 2

Because pi^2/6 is the reciprocal of the probability that two positive integers selected at random are relatively prime.