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Question Number 62412 by mathmax by abdo last updated on 20/Jun/19

calculate  lim_(n→+∞)   ∫_0 ^n  (1−(x/n))^n dx

$${calculate}\:\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx} \\ $$

Commented by mathmax by abdo last updated on 03/Jul/19

let A_n = ∫_0 ^n (1−(x/n))^n dx  ⇒A_n = ∫_R  (1−(x/n))^n  χ_([0,n[) (x)dx  let f_n (x)=(1−(x/n))^n χ_([0,n[) (x)dx   we have f_n  →e^(−x)  on [0,+∞[  and  ∣f_n (x)∣≤ e^(−x)     (theoreme of convergence dominee) give  lim_(n→+∞)  A_n = ∫_0 ^∞

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx}\:\:\Rightarrow{A}_{{n}} =\:\int_{{R}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \:\chi_{\left[\mathrm{0},{n}\left[\right.\right.} \left({x}\right){dx} \\ $$$${let}\:{f}_{{n}} \left({x}\right)=\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \chi_{\left[\mathrm{0},{n}\left[\right.\right.} \left({x}\right){dx}\:\:\:{we}\:{have}\:{f}_{{n}} \:\rightarrow{e}^{−{x}} \:{on}\:\left[\mathrm{0},+\infty\left[\:\:{and}\right.\right. \\ $$$$\mid{f}_{{n}} \left({x}\right)\mid\leqslant\:{e}^{−{x}} \:\:\:\:\left({theoreme}\:{of}\:{convergence}\:{dominee}\right)\:{give} \\ $$$${lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \\ $$

Commented by mathmax by abdo last updated on 03/Jul/19

lim_(n→+∞)  A_n =∫_R lim_(n→+∞)  f_n (x)dx =∫_0 ^∞  e^(−x) dx =[−e^(−x) ]_0 ^(+∞)  =1

$${lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\int_{{R}} {lim}_{{n}\rightarrow+\infty} \:{f}_{{n}} \left({x}\right){dx}\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {dx}\:=\left[−{e}^{−{x}} \right]_{\mathrm{0}} ^{+\infty} \:=\mathrm{1} \\ $$

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