Inspired by Patrick Corn

Calculus Level 5

Let \({ H }_{ n }^{ (2) }=\displaystyle\sum _{ k=1 }^{ n }{ \dfrac { 1 }{ { k }^{ 2 } } } \), and if \[\sum _{ n=1 }^{ \infty }{ \dfrac { { H }_{ n }^{ (2) } }{ { 2 }^{ n } } } \]

is in the form \[\dfrac { { \pi }^{ a } }{ b } -(\ln d)^c, \]

where \(a,b,c\) and \(d\) are integers, find \(a+b+c+d\).


Inspiration.

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