Find the sum of the series 2

$\large \displaystyle S = \sum_{n=0}^{\infty} \frac{\sum_{k=0}^{n} 7^k }{n!}$

$\large \displaystyle S = \sum_{n=0}^{\infty} \frac{ \frac{7^{n+1} -1}{7-1}} {n!}$

$\large \displaystyle S = \frac16 \left [ 7 \sum_{n=0}^{\infty} \frac{7^n}{n!} - \sum_{n=0}^{\infty} \frac{1}{n!} \right ]$

$\color{#20A900} \boxed{\large \displaystyle S = \frac{7e^7 - e}{6}}$

Thus:

$\color{#3D99F6} \large \displaystyle a = 7, b = 6, \boxed{\large \displaystyle a+b=13}$