# 一道求极限题目

$\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\cfrac{e}{e-1}-\sum_{k=1}^{n}{(\cfrac{k}{n})}^{n}\end{smallmatrix}\bigr) =\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{\infty}{e}^{-k}-\sum_{k=0}^{n-1}{(1-\cfrac{k}{n})}^{n}\end{smallmatrix}\bigr) \\=\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{n-1}{e}^{-k}-{(1-\cfrac{k}{n})}^{n}+\sum_{k=n}^{\infty}{e}^{-k}\end{smallmatrix}\bigr) =\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{n-1}{e}^{-k}(1-{(1-\cfrac{k}{n})}^{n}{e}^{k})\end{smallmatrix}\bigr)$$\\=\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{n-1}{e}^{-k}(-ln{(1-\cfrac{k}{n})}^{n}-k+O({(-ln{(1-\cfrac{k}{n})}^{n}-k)}^{2})\end{smallmatrix}\bigr)$$ =\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{n-1}{e}^{-k}(n(\cfrac{k}{n}+\cfrac{{k}^{2}}{2{n}^{2}}+O(\cfrac{{k}^{3}}{3{n}^{3}}))-k+O({(-ln{(1-\cfrac{k}{n})}^{n}-k)}^{2})\end{smallmatrix}\bigr)$$\\=\lim\limits_{n\to \infty}n\bigl(\begin{smallmatrix}\sum_{k=0}^{n-1}{e}^{-k}(\cfrac{{k}^{2}}{2n}+O(\cfrac{{k}^{3}}{{n}^{2}}))+O({(\cfrac{{k}^{2}}{2n})}^{2})\end{smallmatrix}\bigr)$$ =\lim\limits_{n\to \infty}\sum_{k=0}^{n-1}{e}^{-k}\cfrac{{k}^{2}}{2}+\cfrac{O(\sum_{k=0}^{n-1}{e}^{-k}{k}^{3})}{n}+\cfrac{O(\sum_{k=0}^{n-1}{e}^{-k}{k}^{4})}{4n} \\=\lim\limits_{n\to \infty}\sum_{k=0}^{n-1}{e}^{-k}\cfrac{{k}^{2}}{2} =\sum_{k=0}^{\infty}{e}^{-k}\cfrac{{k}^{2}}{2} =S=\sum_{k=1}^{\infty}{e}^{-k+1}\cfrac{{(k-1)}^{2}}{2} \\=eS-\sum_{k=1}^{\infty}{e}^{-k+1}\cfrac{2k-1}{2} =\cfrac{1}{2e-2}\sum_{k=1}^{\infty}{e}^{-k+1}(2k-1) \\=\cfrac{1}{2e-2}\sum_{k=0}^{\infty}{e}^{-k}(2k+1) =\cfrac{1}{2e-2}+{e}^{-1}S+\cfrac{1}{2e-2}\sum_{k=1}^{\infty}2{e}^{-k} \\=\cfrac{1}{1-{e}^{-1}}\cfrac{1}{2e-2}(1+\sum_{k=1}^{\infty}2{e}^{-k}) =\cfrac{{e}^{-1}({e}^{-1}+1)}{2{(1-{e}^{-1})}^{3}}$