sequences and series - Does $sumlimits_{k=1}^n 1 / k ^ 2$ converge when $nrightarrowinfty$?

I can prove this sum has a constant upper bound like this:

$$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1 {k (k - 1)} = 2 - \frac 1 n \lt 2$$

And computer calculation shows that sum seems to converge to 1.6449. But I still want to know:

• Dose this sum converge?


• Is there a name of this sum (or the series $1 / k ^2 $)?

Answer

A sequence that is increasing and bounded must converge. That's one of the fundamental properties of the real line. So once you've observed that your sequence of partial sums is bounded, since it obviously increases, it must converge. Of course it is a very famous series, and it converges to a number which quite miraculously has a "closed form" formula: it is $\pi^2/6$.

EDIT: for many proofs of this famous formula, see this MO question.