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## Is $\int_{0}^{\infty}e^{-x}x^{s}dx \lt \infty$ for $s>-1$ or $s<-1$? [closed]

Hello, I am trying to argue with my friends about the below integral, but I do not know how to go about the explanation.

Somebody told me that $\int_{0}^{\infty}e^{-x}x^{s}dx \lt \infty$ for $s>-1$, but he did not prove it for me.

Two of my friends in our class told me that $\int_{0}^{\infty}e^{-x}x^{s}dx \lt \infty$ for $s<-1$, again without prove.

I am very confused about this, and I really want to know which is which. And also the reason behind it. Can somebody help me out with a detail proof?

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 Perhaps stackexchange would be more helpful. – Igor Rivin May 22 2012 at 20:17 This would be a great question for math.stackexchange.com. – Will Sawin May 22 2012 at 20:17 Why not mathoverflow? – Hamif May 22 2012 at 20:20 mathoverflow.net/faq#whatnot – Yemon Choi May 22 2012 at 20:55