Thursday, April 7, 2016

sequences and series - Weird limit limlimitsnmathoptoinftyfrac1ensumlimitsnkmathop=0fracnkk!




lim




I thought this limit was obviously 1 at first but approximations on Mathematica tells me it's 1/2. Why is this?


Answer



In this answer, it is shown that
\begin{align} e^{-n}\sum_{k=0}^n\frac{n^k}{k!} &=\frac{1}{n!}\int_n^\infty e^{-t}\,t^n\,\mathrm{d}t\\ &=\frac12+\frac{2/3}{\sqrt{2\pi n}}+O(n^{-1}) \end{align}


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