About Euler's formula for Apery number

Question

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 :)

Euler's original is available at your page, also a modern description by Srivastava, or at least a reference. http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant#Series_representation Meanwhile, the best source on special functions is Whittaker and Watson — Will Jagy, Feb 18 '13 at 04:24

score 3 · Accepted Answer · answered Feb 22 '13 at 00:26

3

Some interesting links :

Euler's original paper (E432 link): 'Exercitationes analyticae'
Euler's proof explained (p.1084) in Raymond Ayoub's paper : 'Euler and the zeta function'
A generalization in Srivastava's book 'Series Associated With the Zeta and Related Functions'

Hoping this helped,

answered Feb 22 '13 at 00:26

Raymond Manzoni

44,136

Generalizations are proposed in this later thread. – Raymond Manzoni Feb 09 '14 at 18:33

About Euler's formula for Apery number

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