Menu Close

Prove-that-lim-n-n-1-0-e-nt-1-ln-1-t-dt-2pi-




Question Number 196986 by Erico last updated on 05/Sep/23
Prove that   lim_(n→+∞) n∫^( 1) _( 0) e^(−nt(1−ln(1−t))) dt=(√(2π))
$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}n}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{e}^{−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\right)} \mathrm{dt}=\sqrt{\mathrm{2}\pi} \\ $$
Answered by witcher3 last updated on 07/Sep/23
1−ln(1−t)>1  ⇒−nt(1−ln(1−t)<−nt  ⇔ne^(−nt(1−ln(1−t))) <ne^(−nt)   ∫_0 ^1 ne^(−nt(1−ln(1−t))) <∫_0 ^1 ne^(−nt) <∫_0 ^∞ e^(−t) =1  ⇒lim_(n→∞) n∫_0 ^1 e^(−nt(1−ln(1−t))) dt≤1<(√(2π))  impossibl =(√(2π))
$$\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)>\mathrm{1} \\ $$$$\Rightarrow−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)<−\mathrm{nt}\right. \\ $$$$\Leftrightarrow\mathrm{ne}^{−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\right)} <\mathrm{ne}^{−\mathrm{nt}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ne}^{−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\right)} <\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ne}^{−\mathrm{nt}} <\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{t}} =\mathrm{1} \\ $$$$\Rightarrow\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}n}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{e}^{−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\right)} \mathrm{dt}\leqslant\mathrm{1}<\sqrt{\mathrm{2}\pi} \\ $$$$\mathrm{impossibl}\:=\sqrt{\mathrm{2}\pi} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *