How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers?

Question

I have tested this for specific numbers and it looks like we need to use partial fraction decomposition. Is there some general formula for that we can use here?

In which way is this question connected to [tag:multivariable-calculus]? — José Carlos Santos, Jun 04 '22 at 14:18
It is much easier to show it converges than to compute it. Near $x=0$ compare to $\int x^n$ and near $+\infty$ compare to $\int x^{n-m}$. — GEdgar, Jun 04 '22 at 14:19
Hint: the integrand is asymptotic to $x^n$ ($x^{n-m}$) for small (large) $x$. — J.G., Jun 04 '22 at 14:24
@FShrike i actually made a typo which is edited now. but it looks similar enough to what you linked... — david h, Jun 04 '22 at 14:39
Oh. For that, apply some variable substitutions to arrive at the Beta function, as done on the Wikipedia page — FShrike, Jun 04 '22 at 14:40

score 0 · Answer 1 · answered Jun 04 '22 at 15:45

0

Check that the integral $\int^{\infty} x^{-p} dx$ is finite if $p>1$. So here the given integral is asymptotically $J\sim \int^{\infty} x ^{-(m-n)} dx$ and it will be convergent if $m-n>1$.

answered Jun 04 '22 at 15:45

Z Ahmed

43,235

How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers?

1 Answers1

Linked