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Question Number 193951 by Erico last updated on 23/Jun/23

$$\mathrm{Prove}\mathrm{that}:$$$$\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }\frac{1}{\mathrm{n}!}{\underset{0}{\int }}^{\mathrm{n}}{t}^n{e}^{-t}dt=\frac{1}{2}$$

Answered by senestro last updated on 24/Jun/23

$$\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{n!}{\int }_0^n{t}^n{e}^{-t}dt=\underset{n\to +\mathrm{\infty }}{\lim }-e^{-t}(1+\frac{t}{1!}+\frac{t^2}{2!}+...+\frac{t^n}{n!}){\mid }_0^n$$$$=\underset{n\to +\mathrm{\infty }}{\lim }-e^{-n}(1+\frac{n}{1!}+\frac{n^2}{2!}+...+\frac{n^n}{n!})+\underset{n\to +\mathrm{\infty }}{\lim }{e}^{-0}(1+\frac{0}{1!}+\frac{0^2}{2!}+...+\frac{0^n}{n!})$$$$=\underset{n\to +\mathrm{\infty }}{\lim }-e^{-n}(1+\frac{n}{1!}+\frac{n^2}{2!}+...+\frac{n^n}{n!})+\underset{n\to +\mathrm{\infty }}{\lim 1}\left(1\right)$$$$=\underset{n\to +\mathrm{\infty }}{\lim }-e^{-n}{e}^n+\underset{n\to +\mathrm{\infty }}{\lim 1}$$$$=\underset{n\to +\mathrm{\infty }}{\lim }-1+\underset{n\to +\mathrm{\infty }}{\lim 1}$$$$=-1+1=0$$$$hencethequestioniswrong$$$$\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{n!}{\int }_0^n{t}^n{e}^{-t}dt=0$$$$but$$$$\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{n!}{\int }_0^{\mathrm{\infty }}{t}^n{e}^{-t}dt=1$$$$and$$$$\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{2n!}{\int }_0^{\mathrm{\infty }}{t}^n{e}^{-t}dt=\frac{1}{2}$$$$$$

Answered by aba last updated on 24/Jun/23

$$\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{1}{2\mathrm{n}!}{\int }_0^{\mathrm{\infty }}{\mathrm{t}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}=\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{\mathrm{\Gamma }(\mathrm{n}+1)}{2\mathrm{n}!}=\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{\mathrm{n}!}{2\mathrm{n}!}=\frac{1}{2}✓$$

Commented by Erico last updated on 26/Jun/23

$$\mathrm{Regarde}:$$$$\mathrm{Posons}\mathrm{f}\left(\mathrm{t}\right)=\frac{{\mathrm{t}}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{t}}$$$$\mathrm{\forall }\mathrm{n}\in \mathbb{N}{\underset{\mathrm{n}}{\int }}^{2\mathrm{n}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}={\underset{0}{\int }}^{\mathrm{n}}\mathrm{f}(2\mathrm{n}-\mathrm{t})\mathrm{dt}$$$$\mathrm{or}{\int }_{2\mathrm{n}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}={\underset{\mathrm{n}}{\int }}^{\mathrm{x}-\mathrm{n}}\frac{(\mathrm{u}+\mathrm{n})}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}-\mathrm{n}}\mathrm{du}$$$$\mathrm{or}\mathrm{u}⩾\mathrm{n}\Rightarrow \frac{{(\mathrm{u}+\mathrm{n})}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}-\mathrm{n}}⩽\frac{{\left(2\mathrm{u}\right)}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}-\mathrm{n}}$$$$\Rightarrow \mathrm{Si}\mathrm{x}⩾2\mathrm{n},{\underset{\mathrm{n}}{\int }}^{\mathrm{x}-\mathrm{n}}\frac{{(\mathrm{u}+\mathrm{n})}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}-\mathrm{n}}\mathrm{du}⩽{\left(\frac{2}{\mathrm{e}}\right)}^{\mathrm{n}}{\underset{\mathrm{n}}{\int }}^{\mathrm{x}-\mathrm{n}}\frac{{\mathrm{u}}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}}\mathrm{du}⩽{\left(\frac{2}{\mathrm{e}}\right)}^{\mathrm{n}}{\underset{0}{\int }}^{+\mathrm{\infty }}\frac{{\mathrm{u}}^{\mathrm{n}}}{\mathrm{n}!}{\mathrm{e}}^{-\mathrm{u}}\mathrm{du}$$$$⩽{\left(\frac{2}{\mathrm{e}}\right)}^{\mathrm{n}}$$$$\mathrm{donc}\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }\frac{1}{\mathrm{n}!}{\underset{2\mathrm{n}}{\int }}^{+\mathrm{\infty }}{\mathrm{u}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{u}}\mathrm{du}=0$$$$\Rightarrow \underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }\frac{1}{\mathrm{n}!}{\underset{0}{\int }}^{2\mathrm{n}}{\mathrm{t}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}=1$$$$\Rightarrow \underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }[\frac{1}{\mathrm{n}!}{\underset{0}{\int }}^{\mathrm{n}}{\mathrm{t}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}+\frac{1}{\mathrm{n}!}{\underset{\mathrm{n}}{\int }}^{2\mathrm{n}}{\mathrm{t}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}]=1$$$$\mathrm{donc}\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }\frac{1}{\mathrm{n}!}{\underset{0}{\int }}^{\mathrm{n}}{\mathrm{t}}^{\mathrm{n}}{\mathrm{e}}^{-\mathrm{t}}\mathrm{dt}\ne 0$$$$$$

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