Benedict Irwin edited Explaination.tex  over 9 years ago

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\int_0^\infty x^n\frac{d^n}{dx^n}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\;dx=(-1)^nn!\zeta(s)\Gamma(s)+\sum_{k=1}^n\frac{(-1)^{k+n}n!}{k!}\bigg[x^k\frac{d^{k-1}}{dx^{k-1}}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\bigg]_0^\infty  \end{equation}  Or rearranging \begin{equation}  \zeta(s)=\frac{(-1)^n}{n!\Gamma(s)}\Bigg[\int_0^\infty x^n\frac{d^n}{dx^n}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\;dx-\sum_{k=1}^n\frac{(-1)^{k+n}n!}{k!}\bigg[x^k\frac{d^{k-1}}{dx^{k-1}}\bigg(\frac{x^{s-1}}{e^x-1}\bigg)\bigg]_0^\infty\Bigg]  \end{equation}