# How to solve the following limit?

$$\lim_{n\to\infty}{\bigg(1-\cfrac{1}{2^2}\bigg)\bigg(1-\cfrac{1}{3^2}\bigg) \cdots \bigg(1-\cfrac{1}{n^2}\bigg)}$$

This simplifies to $\prod_{n=1}^{\infty}{\cfrac{n(n+2)}{(n+1)^2}}$. Besides partial fractions and telescope, how else can we solve? Thank you!

-
Using the fact $x^2-y^2=(x-y)(x+y)$ –  Jlamprong Feb 3 '14 at 11:00
$\prod_{n=2}^{\infty}{\cfrac{(n-1)(n+1)}{n^2}}$. A further hint? –  Daniel C Feb 3 '14 at 11:04
I suppose there are typo's in your product formula. –  Claude Leibovici Feb 3 '14 at 11:08
Yes. Expanding the product it's the key to observe that they simplify each other. Thank you! –  Daniel C Feb 3 '14 at 11:15
See Basel problem. –  Lucian Feb 3 '14 at 13:02