Not ordinary sum-1

Calculus Level 4

\[\large \sum _{ n=1 }^{ \infty }{ \dfrac { { n }^{ 3} }{ { 3 }^{ n }\cdot n! } } =\dfrac { c~\sqrt [ a ]{ e } ~\pi^{d-1}}{ b } \]

If the equation above holds true for positive integers \(a\), \(b\), \(c\), and \(d\), while \(c\) and \(b\) are coprime, find \(a+b+c+d\).

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


Inspiration.

Also try part-2 here.

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