Sum of Gaps

Calculus Level 3

Given that \(\displaystyle e = \sum_{k=0}^\infty \dfrac1{k!} \approx 2.718, \) what is the value of \[ \displaystyle \sum_{n=0}^{\infty}\left(e- \sum_{k=0}^{n} \frac{1}{k!}\right) ?\]