About Euler's formula for Apery number

Euler's formula. $$\zeta(3)=\frac{\pi^2}{7}\left(1-4\sum_{m\ge 1}\frac{\zeta(2m)}{(2m+1)(2m+2)2^{2m}}\right)$$
I saw this formula in Wikipedia a few months ago. I have searched about Euler's original proof for the formula, but I couldn't find any useful information. So I just tried to prove the formula on my own, and I posted my post in here.

But I doubt that Euler's skills used in his proof is similar to mine. I believe that he could think more clever skills which led him to discovery of the formula.

Could you provide some information about Euler's original proof? Or, Do you have other kinds of proof for the formula? Thanks :)

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Euler's original is available at your page, also a modern description by Srivastava, or at least a reference. en.wikipedia.org/wiki/… Meanwhile, the best source on special functions is Whittaker and Watson –  Will Jagy Feb 18 '13 at 4:24