'e'eesh! What a series!

Calculus Level 4

The sum of this series:

\[ \displaystyle \sum_{n = 0}^{\infty} \dfrac 1 {n!} \left[ \sum_{k=0}^n \left( \dfrac {k-1} {\ln 2} + \left(k+1\right) \int_0^1 2^{-(k+1)x} dx \right) \right] \]

can be expressed in the form:

\[ \displaystyle \dfrac{ e^a + (\sqrt{e})^b } { \ln c }, \]

where \(a, b, c \in \mathbb{N} \). What is the value of \(a + b + c \) ?

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