convergence of an integral

Quote:

Originally Posted by Random Variable

Does $\int_{0}^{\infty} \frac{\ln^{n} x}{1+x^{2}} \ dx$ converge for all $n \in \mathbb{N}$ ?

If it does, it's not too hard to show that it equals $0$ when $n$ is odd and $2n! \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^{n+1}}$ when $n$ is even.

Is $x^{2}$ always eventually going to grow much faster than $\ln^{n} x$ no matter how large $n$ is ?

As $x\to\infty$ , $\ln x$ increases more slowly than any positive power of x. Thus $\ln x \ll x^{1/(2n)}$ and therefore $\frac{\ln^nx}{x^2} < \frac1{x^{3/2}},$ which is enough to ensure convergence of your integral at ∞.

There is also a singularity at 0. But near 0 the integral looks like $\int_0\ln^nx\,dx$ , and that also converges.