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Question Number 196986 by Erico last updated on 05/Sep/23

$$\mathrm{Prove}\mathrm{that}$$$$\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim n}{\underset{0}{\int }}^1{\mathrm{e}}^{-\mathrm{nt}(1-\ln (1-\mathrm{t}))}\mathrm{dt}=\sqrt{2\pi }$$

Answered by witcher3 last updated on 07/Sep/23

$$1-\ln (1-\mathrm{t})>1$$$$\Rightarrow -\mathrm{nt}(1-\ln (1-\mathrm{t})<-\mathrm{nt}$$$$\iff {\mathrm{ne}}^{-\mathrm{nt}(1-\ln (1-\mathrm{t}))}<{\mathrm{ne}}^{-\mathrm{nt}}$$$${\int }_0^1{\mathrm{ne}}^{-\mathrm{nt}(1-\ln (1-\mathrm{t}))}<{\int }_0^1{\mathrm{ne}}^{-\mathrm{nt}}<{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{t}}=1$$$$\Rightarrow \underset{\mathrm{n}\to \mathrm{\infty }}{\lim n}{\int }_0^1{\mathrm{e}}^{-\mathrm{nt}(1-\ln (1-\mathrm{t}))}\mathrm{dt}⩽1<\sqrt{2\pi }$$$$\mathrm{impossibl}=\sqrt{2\pi }$$

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