Calculate the limit of the sequence
$\lim_{n\rightarrow\infty}\ a_n$
$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right), n\geqslant2 $
Here is what I did:
$\left(1-\frac{1}{n^2}\right)=\left[\left(1-\frac{1}{n^2}\right)^{n^2}\right]^\frac{1}{n^2}=e^\frac{1}{n^2}$
$a_n=e^{\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}}$
$\Rightarrow \lim_{n\rightarrow\infty}\ a_n=e^{\frac{1}{\infty}}=e^0=1$
Not sure if I'm on the right track..
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