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Does this product converge? $\prod_{n=1}^{\infty}\frac{1}{n^{2}+1}$

any hint?

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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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Your hint is: It's the product of a lot of numbers each of which is between 0 and 1.

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    @mathst., I would have pointed out that all the numbers being multiplied are positive and no larger than $1/2.$2012-10-01