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

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

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