Summation Contest Conceptual Problem On Digamma Function

Calculus Level 5

\[\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 2 }+4 } } =\frac { \pi }{ A } \coth { \left( B\pi \right) } -\frac { 1 }{ C } \]

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

Hint: Exploit partial fractions and relate it to the digamma function.

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