<s>[INST] What is the limit of $\lim_{n\to \infty} (1 - \frac{1}{4})(1 - \frac{1}{9})(1 - \frac{1}{16}) \cdots (1 - \frac{1}{(n+1)^2})$? What is the evaluation of the following infinite series?
$$\lim_{n\to \infty} \left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right)\left(1 - \frac{1}{16}\right) \cdots \left(1- \frac{1}{(n+1)^2}\right)$$
I've tried to simplify each expression which left me with:
$$\lim_{n\to \infty} \frac{3\times8\times15\times24\times\cdots\times((n+1)^2-1)}{4\times9\times16\times25\times\cdots\times(n+1)^2}$$
Is this a good way to approach this problem? 
 [/INST] hint
Take logarithm and use
$$
\sum_{k=2}^{n+1}\ln\left(1-\frac{1}{k^2}\right)
= \sum_{k=2}^{n+1}\left(\ln\left(\frac{k-1}{k}\right)+\ln\left(\frac{k+1}{k}\right)\right)
=\ln\left(\frac{1}{n+1}\right)+\ln\left(\frac{n+2}{2}\right).
$$
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