- 22

- 0

**1. Homework Statement**

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

**2. Homework 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.