Solve if You Can

Calculus Level pending

Given

\[ S = 1 + \frac{ \sqrt{2 } - 1 } { 2 \sqrt{2} } + \frac{ 3 - 2 \sqrt{2} } { 12} + \frac{ 5 \sqrt{2} - 7 } { 24 \sqrt{2} } + \frac{ 17 - 12 \sqrt{2} } {80 } + \ldots ,\]