Too much of summation!

Calculus Level 5

Find the value of:

\[\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { \left( -1 \right) }^{ k } }{ { \left( k+1 \right) }^{ 4s-1 } } \right) } \right) } -1 \right) } \]

The answer is of the form:

\[\frac { A }{ B } -\frac { \pi }{ C } \left( \frac { { e }^{ D\pi }+F }{ { e }^{ E\pi }-G } \right) \]

Find: \(A+B+C+D+E+F\)

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