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?
