(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show [tex]\mathop{\lim}\limits_{n \to \infty}(\frac{1}{n!}\int_{1}^{\infty}x^n\frac{1}{e^x} dx )=1[/tex]

2. Relevant equations

The hint is that [tex]e=\mathop{\lim}\limits_{n \to \infty}\sum_{k=0}^{n}1/k![/tex]

3. The attempt at a solution

First I wrote out the improper integral as limit of a proper integral. Then I tried to integrate by parts with u=x^n dv=e^-xdx...which eventually gets that proper integral to be [tex]1/e-b^n/e^b+n\int_{1}^{b}\frac{1}{e^x}x^(n-1)dx[/tex]

But even when I take limit b->infinity of this remaining integral either it's going to exist and be finite in which case when i take n->infinity it will be infinite or else it dosn't exist or is not finite in which case the original integral is not improperly integral. I tried using u=e^-x and dv=x^n with little success as well--did I make a mistake somewhere here?

Thanks.

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# Homework Help: Prove limit of improper Integral is 1

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