Hi, trying to complete Problem Set 8B, Part II. I can do problem 1a use L'Hopital to evaluate lim as x - inf of (x^m) exp(-x) = 0. But can't figure out how to use that in part b, along with "limit comparison" to show that integral from 0 to inf of (x^n)exp(-x)dx converges for n = 0. I think the professor said in a lecture that just because the integrand goes to 0 at inf, doesn't prove that such an integral converges. Can anyone help? Thanks!

Question

Hi, trying to complete Problem Set 8B, Part II. I can do problem 1a use L'Hopital to evaluate lim as x -> inf of (x^m) exp(-x) = 0. But can't figure out how to use that in part b, along with "limit comparison" to show that integral from 0 to inf of (x^n)exp(-x)dx converges for n >= 0. I think the professor said in a lecture that just because the integrand goes to 0 at inf, doesn't prove that such an integral converges. Can anyone help? Thanks!

phi · Answer

See the last page of http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/lec35.pdf