Summation Contest Conceptual Problem On Digamma Function

Calculus Level 5

n=11n2+4=πAcoth(Bπ)1C\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 AA, BB and CC. Find A+B+CA+B+C.

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

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