Integral Stuff 5

$\begin{aligned} I & = \int_1^\infty \frac {e^\frac 1x}{x^{16}}\ dx & \small \color{#3D99F6} \text{Let } u = \frac 1x \implies du = - \frac 1{x^2} \ dx \\ & = \int_0^1 u^{14}e^u \ du & \small \color{#3D99F6} \text{By integration by parts.} \\ & = u^{14}e^u - 14 u^{13}e^u + 14\cdot 13 u^{12}e^u + \cdots + \frac {14!}{2!} u^2e^u - \frac {14!}{1!} u e^u + \frac {14!}{0!} e^u \bigg|_0^1 \\ & = e\sum_{n=0}^{14} \frac {(-1)^n14!}{n!} - 14! \\ & = 32071101049e - 87178291200 \end{aligned}$

$\implies a+b = - 87178291200 + 32071101049 = \boxed{-55107190151}$