Question Number 192276 by York12 last updated on 13/May/23
![lim_(n→∞) (Σ_(k=0) ^n [((k(n−k)!+(k+1))/((k+1)!(n−k)!))])](https://www.tinkutara.com/question/Q192276.png)
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left[\frac{{k}\left({n}−{k}\right)!+\left({k}+\mathrm{1}\right)}{\left({k}+\mathrm{1}\right)!\left({n}−{k}\right)!}\right]\right) \\ $$
Answered by witcher3 last updated on 14/May/23
$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}!\left(\mathrm{n}−\mathrm{k}\right)!}=\frac{\mathrm{1}}{\mathrm{n}!}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}!}{\mathrm{k}!\left(\mathrm{n}−\mathrm{k}\right)!} \\ $$$$=\frac{\mathrm{1}}{\mathrm{n}!}.\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} =\frac{\mathrm{2}^{\mathrm{n}} }{\mathrm{n}!} \\ $$$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{k}\left(\mathrm{n}−\boldsymbol{\mathrm{k}}\right)!+\left(\mathrm{k}+\mathrm{1}\right)}{\left(\mathrm{k}+\mathrm{1}\right)!\left(\mathrm{n}−\mathrm{k}\right)!}\right)=\Sigma\frac{\mathrm{1}}{\mathrm{k}!}+\frac{\mathrm{1}}{\mathrm{k}!\left(\mathrm{n}−\mathrm{k}\right)!} \\ $$$$=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}!}+\frac{\mathrm{2}^{\mathrm{n}} }{\mathrm{n}!} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}=\mathrm{e} \\ $$$$ \\ $$
Commented by York12 last updated on 20/Aug/23
![lim_(n→∞) (Σ_(k=1) ^n [((k(n−k)!+(k+1))/((k+1)!(n−k)!))]) = lim_(n→∞) (Σ_(k=1) ^n [(k/((k+1)!))+(1/(k!(n−k)!))]) =lim_(n→∞) (Σ_(k=1) ^n [(1/((k!)))−(1/((k+1)!))])+lim_(n→∞) ((2^n /(n!))) = lim_(n→0) (1−(1/((n+1)!))+0)=1 → (That′s it )](https://www.tinkutara.com/question/Q192320.png)
$$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{{k}\left({n}−{k}\right)!+\left({k}+\mathrm{1}\right)}{\left({k}+\mathrm{1}\right)!\left({n}−{k}\right)!}\right]\right)\:=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{{k}}{\left({k}+\mathrm{1}\right)!}+\frac{\mathrm{1}}{\mathrm{k}!\left({n}−{k}\right)!}\right]\right) \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{\mathrm{1}}{\left({k}!\right)}−\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)!}\right]\right)+\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{2}^{{n}} }{{n}!}\right)\:=\:\underset{{n}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{1}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!}+\mathrm{0}\right)=\mathrm{1}\:\:\rightarrow\:\left({That}'{s}\:{it}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$