Compute the cool integral/sum

$\begin{aligned} X & = \ln \left(\sum_{\alpha =1}^\infty \int_0^\infty \frac {x^{\alpha-1}e^{-\frac x{2020}}}{2020((\alpha-1)!)^2} dx \right) & \small \blue{\text{Let }n = \alpha -1} \\ & = \ln \left(\sum_{n=0}^\infty \int_0^\infty \frac {x^n e^{-\frac x{2020}}}{2020(n!)} dx \right) & \small \blue{\text{Let }t = \frac x{2020} \implies dt = \frac {dx}{2020}} \\ & = \ln \left(\sum_{n=0}^\infty \frac {2020^n}{(n!)^2} \blue{\int_0^\infty t^n e^{-t}\ dt}\right) \\ & = \ln \left(\sum_{n=0}^\infty \frac {2020^n}{(n!)^2} \cdot \blue{\Gamma(n+1)}\right) & \small \blue{\text{where }\Gamma(\cdot) \text{ is the gamma function.}} \\ & = \ln \left(\sum_{n=0}^\infty \frac {2020^n}{(n!)^2} \cdot \blue{n!}\right) \\ & = \ln \left(\sum_{n=0}^\infty \frac {2020^n}{n!} \right) \\ & = \ln \left(e^{2020} \right) \\ & = \boxed{2020} \end{aligned}$

Reference: Gamma function