lim_( n→∞) (∫_0 ^( 1) (((1−x)^n e^x )/(n!))dx)=?


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Question Number 138801 by mnjuly1970 last updated on 18/Apr/21

${l i m}_{n \to \infty} (\int_{0}^{1} \frac{{(1 - x)}^{n} e^{x}}{n!} d x) = ?$
	Answered by Kamel last updated on 18/Apr/21

	Commented by mnjuly1970 last updated on 18/Apr/21

	$t h a n k s m r k a m e l$ $v e r y n i c e . . .$
	Answered by mr W last updated on 18/Apr/21

	$\int_{0}^{1} 0 d x ⩽ \int_{0}^{1} \frac{{(1 - x)}^{n} e^{x}}{n!} d x ⩽ \int_{0}^{1} \frac{e^{x}}{n!} d x$ $0 ⩽ \int_{0}^{1} \frac{{(1 - x)}^{n} e^{x}}{n!} d x ⩽ \frac{e - 1}{n!}$ $0 ⩽ \lim_{n \to \infty} \int_{0}^{1} \frac{{(1 - x)}^{n} e^{x}}{n!} d x ⩽ 0$ $\Rightarrow \lim_{n \to \infty} \int_{0}^{1} \frac{{(1 - x)}^{n} e^{x}}{n!} d x = 0$
	Commented by mnjuly1970 last updated on 18/Apr/21

	$g r a t e f u l m r W . . .$
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