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

\underset{n \to + \infty}{\lim n} {\int_{0}}^{1} e^{- nt (1 - \ln (1 - t))} dt = \sqrt{2 π}

Answered by witcher3 last updated on 07/Sep/23

1 - \ln (1 - t) > 1

\Rightarrow - nt (1 - \ln (1 - t) < - nt

\Leftrightarrow {ne}^{- nt (1 - \ln (1 - t))} < {ne}^{- nt}

\int_{0}^{1} {ne}^{- nt (1 - \ln (1 - t))} < \int_{0}^{1} {ne}^{- nt} < \int_{0}^{\infty} e^{- t} = 1

\Rightarrow \underset{n \to \infty}{\lim n} \int_{0}^{1} e^{- nt (1 - \ln (1 - t))} dt ⩽ 1 < \sqrt{2 π}

impossibl = \sqrt{2 π}

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