The infinite series,

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?