sandu2508 wrote: Let $ a_n = 1 + \dfrac1{2^2} + \dfrac1{3^2} + \cdots + \dfrac1{n^2}$ Find $ \lim_{n\to\infty}a_n$ How come this be an MO problem? I thought this was a known fact. (Or Correct me if I'm wrong. ) $ \lim_{n\to\infty}a_n = \frac{\pi^2}{6}$ However, I don't know the proof. I've heard that it uses analytic knowledge.

epsilonist wrote: sandu2508 wrote: Let $ a_n = 1 + \dfrac1{2^2} + \dfrac1{3^2} + \cdots + \dfrac1{n^2}$ Find $ \lim_{n\to\infty}a_n$ How come this be an MO problem? I thought this was a known fact. (Or Correct me if I'm wrong. ) $ \lim_{n\to\infty}a_n = \frac {\pi^2}{6}$ However, I don't know the proof. I've heard that it uses analytic knowledge. A proof can be this: Step 1: Proof that $ \sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^2}{\pi^2 n^2}\right)$ Step 2: See the expansion of $ \sin z /z$ in powers of $ z$, Compare the coefficent of $ z^2$ in the expansion of $ \sin z /z$ with the coefficent in the product of the same power of $ z$.