Is called the Basel Problem.
It was first solve (non-rigorusly) by Euler.
Basically what this says is that the sequence of partial sums,
Has a least upper bound which is .
Thus, if represents the n-th partial sum we have (since it is an upper bound) that,
Subtract from both sides,
Which is true for .
Note, if you wish I can show you how to prove the Basel sum using a Fourier series?