求极限 lim(n→∞)∑(k=1,∞)[1／(k+1)!+1／(k+2)!+1／(k+3)!+…+1／(k+n)!]

zhang@math

新人拜求
\(\lim_{n\to\infty}\left\{ \sum_{k=0}^{\infty}\frac{1}{\left( k+1\right)!}+\sum_{k=0}^{\infty}\frac{1}{\left( k+2\right)!}+\sum_{k=0}^{\infty}\frac{1}{\left( k+3\right)!}+\cdots+\sum_{k=0}^{\infty}\frac{1}{\left( k+n\right)!}\right\}\)

cgl_74

答案为e。
思路是把上面的和式，重新整理为一个和式（矩阵的一半，从按行求和变换为按列求和）。和式可以分为2部分。
再利用指数级数展开式求和。一部分收敛于e，另一部分和式收敛于0.

王守恩

答案为e。

\(\displaystyle\lim_{n\to\infty}\bigg\{\sum_{k=0}^{\infty}\frac{1}{(k+1)!}+\sum_{k=0}^{\infty}\frac{1}{(k+2)!}+\sum_{k=0}^{\infty}\frac{1}{(k+3)!}+…+\sum_{k=0}^{\infty}\frac{1}{(k+n)!}\bigg\}\)
\(=\frac{1}{1!}+\frac{2}{2!}+\frac{3}{3!}+\frac{4}{4!}+\frac{5}{5!}+\frac{6}{6!}+\frac{7}{7!}+…+\frac{n}{n!}\)
\(=\frac{1}{1!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+…+\frac{1}{(n-1)!}\)
\(=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+…+\frac{1}{(n-1)!}\)
\(\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}=e\)

注意下面这些算式之间的区别。

\(1,\displaystyle\sum_{k=0}^{\infty}\frac{1+a}{k!}\ \ \ \ a=0,1,2,3,4,5,6,7,8,9,...\)
     \(e,\ 2e,\ 3e,\ 4e,\ 5e,\ 6e,\ 7e,\ 8e,\ 9e,\ 10e, ...\)

\(2,\displaystyle\sum_{k=0}^{\infty}\frac{k+a}{k!}\ \ \ \ a=0,1,2,3,4,5,6,7,8,9,...\)
     \(e,\ 2e,\ 3e,\ 4e,\ 5e,\ 6e,\ 7e,\ 8e,\ 9e,\ 10e ...\)

\(3,\displaystyle\sum_{k=1}^{\infty}\frac{k+a}{k!}\ \ \ \ a=0,1,2,3,4,5,6,7,8,9,...\)
     \(e,2e-1,3e-2,4e-3,5e-4,6e-5,...\)

\(4,\displaystyle\sum_{k=1}^{\infty}\frac{k-a}{k!}\ \ \ \ a=0,1,2,3,4,5,6,7,8,9,...\)
     \(1,\ 2-e,\ 3-2e,4-3e,5-4e,6-5e,...\)

\(5,\displaystyle\sum_{k=0}^{\infty}\frac{k-a}{k!}\ \ \ \ a=0,1,2,3,4,5,6,7,8,9,...\)
     \(e,0,-e,-2e,-3e,-4e,-5e,-6e,-7e, ...\)
.............

luyuanhong