$$\zeta \left(s\right)=\frac{1}{1-2^{1-s}}\sum _{n=0}^{\infty }\frac{1}{2^{n+1}}\sum _{k=0}^n{\left(-1\right)}^k\left(\begin{array}{c}n\\ k\end{array}\right){\left(k+1\right)}^{-s}$$