Summation Contest Conceptual Problem-4

Calculus Level 5

\[\sum _{ n=1 }^{ \infty }{ \dfrac { 1 }{ { ( { n }^{ 2 }+4 ) }^{ 2 } } } =\dfrac { 1 }{ A } \left ( -1+\pi \coth { ( B\pi ) } +C{ \pi }^{ D }{ \left( \text{csch}\left( E\pi \right) \right) }^{ F } \right) \]

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

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


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