Easy Hyperbolic Summation

Calculus Level 4

\[\large{S= \sum_{n=1}^\infty \dfrac1{(2n-1)2^{n-1}} = 1+\frac { 1 }{ 6 } +\frac { 1 }{ 20 } +\frac { 1 }{ 56 } +\cdots}\]

If \(S\) can be represented as \(\sqrt { A } \sinh ^{ -1 }{ ( B ) } \), where \(A\) and \(B\) are integers . Find \( 100(A+B) \).

Bonus: Find the partial sum of the series above.

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