Checking the convergence of this improper integral $\int_{-\infty}^{\infty} x^ne^{-|x|}dx$

Question

need to decide rather the this integral converges or not: $$\int_{-\infty}^{\infty} x^ne^{-|x|}dx$$

is it possible to say that it converges beacuse it's "tail"->0 and the function itself is continous and blocked?

Answer 1

That $x^n\mathrm e^{-|x|}\to0$ is true but not sufficient to prove the convergence of the integral (otherwise the integral of $1/x$ would converge at $+\infty$). Rather, one should bound $|x^n\mathrm e^{-|x|}|$ by an integrable function.

Hint: For every $n\geqslant0$, there exists some $C_n$ such that $|x^n\mathrm e^{-|x|}|\leqslant C_n\mathrm e^{-|x|/2}$ for every $x$.

To show this, start with the case $n=1$, showing that $C_1=2$ works, then deduce the general case from this case (for $n\geqslant1$, the value $C_n=(2n)^n$ might be enough).