The user known as sos440 posted this: $\begin{align*} \sum_{n=0}^\infty \frac{r^n}{n!} \int_0^\infty x^n e^{-x} \; dx & = \int_{0}^\infty \sum_{n=0}^\infty \frac{(rx)^n}{n!} e^{-x} \; dx = \int_0^\infty e^{-(1 - r)x} \; dx \\ & = \frac{1}{1 - r} = \sum_{n=0}^\infty r^n \end{align*}$ (citing Tonelli's theorem to justify interchanging the sum and the integral), and concluded that $ \frac{1}{n!}\int_0^\infty x^n e^{-x}\,dx=1. $
Is this a very isolated thing or just an instance of something generally useful for some much broader class of integrals? (Obviously, what is seen below is generally true, but how generally is it useful?)
$ \sum_{n=0}^\infty r^n \int_A f_n(x)\,dx = \int_A \sum_{n=0}^\infty \big( r^nf_n(x) \big) \, dx = \int_A g(r,x) \, dx = h(r) = \sum_{n=0}^\infty c_n r^n $ $ \therefore \int_A f_n(x)\,dx = c_n. $