Trigamma summation

Calculus Level 5

\[\large \sum _{ n=1 }^{ \infty }{ \dfrac { { \psi }_{ 1 }( n ) }{ { 2 }^{ n-1 } } } =\dfrac { { \pi }^{ A } }{ B } +{ ( \ln { C } ) }^{ D }\]

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

Notation: \( \psi_1 (\cdot) \) denotes the \(1^\text{st} \) derivative of the Digamma function.

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