# Convergence of $$\int_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x$$

The question for what values of $$p$$ does the improper integral

\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x

converge was asked at Mathematics Stack Exchange.

My solution was to note that this integral can be expressed in terms of the upper incomplete Gamma function

\Gamma(a,x) = \int\limits_{x}^{\infty} \mathrm{e}^{-z} z^{a-1} \mathrm{d} z

Thus

\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x = \Gamma(1+p,1)

As noted here, the upper incomplete Gamma function, $$\Gamma(a,x)$$ is an entire
function for all $$a$$ when $$x \ne 0$$. Thus the integral converges for all values
of $$p$$.