It is depends on your definition of convergence of the product. It is not uncommon to say that this product _diverges_ to zero. This is because the sum $\sum_{n=1}^\infty \log\left(\frac{1}{1+n^2}\right)$ is divergent. – 2012-10-01
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I was confused about this fact: The product of positive real numbers$ a_n<1 $, $ \prod_{n=1}^{\infty} a_n$ converges if and only if the sum $\sum_{n=1}^{\infty} \log a_n$ converges. So how this doesn't contradicts MJD answer below? (btw, by converge I mean finite) – 2012-10-01
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