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{\displaystyle \sum _{k=1}^{\infty }\left({\frac {k^{k}}{e^{k}\,k!}}-{\frac {1}{\sqrt {2\pi k}}}\right)=-{\frac {2}{3}}-{\frac {1}{\sqrt {2\pi }}}\,\zeta \left({\frac {1}{2}}\right)}
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