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Question Number 28430 by abdo imad last updated on 25/Jan/18

$$letgive{A}_n={\int }_0^n{(1+\frac{x}{n})}^n{e}^{-2x}dx{lim}_{n\to \propto }{A}_n?$$$$$$

Commented by abdo imad last updated on 27/Jan/18

$$wehave{(1+\frac{x}{n})}^n=e_{n\to +\mathrm{\infty }}^{nln(1+\frac{x}{n})}\to e^{x}so$$$$A_n={\int }_R{(1+\frac{x}{n})}^n{e}^{-2x}{\chi }_{[0,n[}\left(x\right)dx={\int }_R{f}_n\left(x\right)dx$$$$but{f}_n\left(x\right)\to e^{-x}on[0,+\mathrm{\infty }[so{lim}_{n\to +\mathrm{\infty }}{A}_n$$$$={\int }_0^{\mathrm{\infty }}{e}^{-x}dx={[-e^{-x}]}_0^{x\to +\mathrm{\infty }}=1$$

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