Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function

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Theorem

Let $\zeta$ be the Riemann zeta function.

Let $s\in\C$ be a complex number with real part $\Re(s)>1$.


Then: $\displaystyle \zeta \left({s}\right) = \frac 1 {1 - 2^{1-s} } \sum_{n \mathop = 0}^\infty \left({\frac 1 {2^{n+1} } \sum_{k \mathop = 0}^n \left({-1}\right)^k \binom n k \left({k + 1}\right)^{-s} }\right)$


Proof

Use Riemann Zeta Function in terms of Dirichlet Eta Function and Binomial Theorem.