vasorasy8

2022-07-10

Is there a general formula for
${I}_{n}={\int }_{0}^{1}\left(x-1\right)\left(x-2\right)\dots \left(x-n\right)dx$

behk0

Expert

In terms of the Stirling numbers of the first kind,
$\left(x-1\right)\left(x-2\right)\cdots \left(x-n\right)=\sum _{k=0}^{n}s\left(n,k\right)\left(x-1{\right)}^{k}.$
Thus,
${I}_{n}=\sum _{k=0}^{n}\left(-1{\right)}^{k}\frac{s\left(n,k\right)}{k+1}.$
Asymptotically,
${I}_{n}\sim \left(-1{\right)}^{n}\frac{n!}{\mathrm{log}n}$
as $n\to +\mathrm{\infty }$

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