How Can the Infinite Product of (1+n^-2) Be Bounded?

[SumThePrimes]

I was wondering about the product n=1 to infinity of (1+n^-2) , I used a very unorthodox, to say the least, manipulation of complex numbers that shows that it should equal a particular number larger than 10^17 , however wolfram alpha can't seem to give me a answer, and when I sum to very large numbers, I never get over 4, although this maybe due to rounding errors. I would like any significant bounds on it whatsoever, I know it is greater than three and I know that it does converge... I was thinking of comparing it to series larger or smaller and then bounding it from above or below, but that's obvious, I think... I thought about a partial sum formula with a limit, I got 2(n!) in the denominator and in the numerator I was clueless, again wolfram alpha gave a crazy formula for the mth partial sum, I don't understand that. I assume I am wrong as 10^17 is so large for this series, but would like to be sure; or have a chance.

 

[l'Hôpital]

http://en.wikipedia.org/wiki/Infinite_product#Product_representations_of_functions

Look at the one for sine. What happens if you plug in z = i?

 

[SumThePrimes]

Wow, thank you, I am acquainted with the sines of imaginary numbers and that formula, but it never even crossed my mind... So much for e^(4*pi^2)-1 ... I wouldn't have thought an exact solution... Maybe Euler is better at infinite products than me(Just maybe... ), I actually got this from a self-derived product ... It still has merit if it converges everywhere I guess... Better than my last one that only registered ∞ or 0... so close...