Convergence of \(\int_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x\)

The question for what values of \(p\) does the improper integral
\begin{equation}
\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x
\end{equation}
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
\begin{equation}
\Gamma(a,x) = \int\limits_{x}^{\infty} \mathrm{e}^{-z} z^{a-1} \mathrm{d} z
\end{equation}
Thus
\begin{equation}
\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x = \Gamma(1+p,1)
\end{equation}
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 \(p\).

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