For what values is this integral convergent?

Question

How can I find for what values of $r$ $$\int_0^\infty x^re^{-x}dx$$ converges?

I started by rewriting it as $$\lim_{b\to\infty}\int_0^bx^re^{-x}dx$$ but am not sure how to figure it out from here.

Answer 1

We have

$$x^re^{-x}\sim_0\frac{1}{x^{-r}}\in L^1(0,1)\iff -r<1\iff r>-1$$ and

$$x^re^{-x}=_\infty o\left(\frac1{x^2}\right)\in L^1(1,\infty)$$ We conclude that the given integral exists if and only if $r>-1$.