<s>[INST] What is the limit of the sequence $a_n=(1-\frac{1}{2^2})(1-\frac{1}{3^2})(1-\frac{1}{4^2})\cdots(1-\frac{1}{n^2})$? I am stuck on finding the limit of
$$\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\cdots\left(1-\frac{1}{n^2}\right)$$
Can anybody help?
UPDATE
Is there a solution involving making the fractions like
$$(\frac{n^2-1}{n^2})$$
and using the formula
$$(2^23^24^2...n^2) = (n!)^2$$?
 [/INST] Hint: use Euler's formula
$$\frac{\sin(\pi z)}{\pi z} = (1-z^2/1^2)(1-z^2/2^2)(1-z^2/3^2)\dots$$
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