Sum of Gaps

Calculus Level 2

Given that e=k=01k!2.718,\displaystyle e = \sum_{k=0}^\infty \dfrac1{k!} \approx 2.718, what is the value of n=0(ek=0n1k!)? \displaystyle \sum_{n=0}^{\infty}\left(e- \sum_{k=0}^{n} \frac{1}{k!}\right) ?

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