But how?

Calculus Level 5

\[ \large \displaystyle\sum_{n=1}^\infty\sum_{k=1}^\infty\sum _{ i=1 }^{ \infty }{ \dfrac { nki }{ (n+k+i)! } } =\dfrac { A }{ B } e\]

The equation above holds true for coprime positive integers \(A\) and \(B\). Find \(A+B\).

Clarification: \(e \approx 2.71828\) is the Euler's number.

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