Apéry's Constant in terms of Central Binomial Coefficients
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Theorem
Apéry's constant can be expressed in terms of binomial coefficients as:
- $\map \zeta 3 = \ds \dfrac 5 2 \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n - 1} } {n^3 \dbinom {2 n} n}$
Proof
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Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,20205 69 \ldots$
- Weisstein, Eric W. "Apéry's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AperysConstant.html