Solve if You Can

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We can see here that ${ ❤ }{ ❤ }$ $(\sqrt{2} - 1)^{2} = 3 - 2\sqrt{2} , (\sqrt{2} - 1)^{3} = 5\sqrt{2} - 7 , (\sqrt{2} - 1)^{4} = 17 - 12\sqrt{2} ..$

Let $\dfrac{\sqrt{2} - 1}{\sqrt{2}} = x$

Thus,

$S = 1 + \dfrac{x}{2} + \dfrac{x^{2}}{6} + \dfrac{x^{3}}{12} ........$

$\boxed{S = 1 + \dfrac{x}{1 \times 2} + \dfrac{x^{2}}{2\times3} + \dfrac{x^{3}}{3\times4} ................}$

$S = 1 + ( 1 - \frac{1}{2})x + (\frac{1}{2} - \frac{1}{3})x^{2} + (\frac{1}{3} - \frac{1}{4})x^{3} + ..............$

$S = ( 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + .....) - (\frac{x}{2} + \frac{x^{2}}{3} + \frac{x^{3}}{4} + ....$

$S = 1 - log(1 - x) - \frac{1}{x}(\frac{x^{2}}{2} + \frac{x^{3}}{3} + .........)$

$S = 1 - log(1- x) - \frac{1}{x}( -log(1 - x) -x)$

$S = 2 - \frac{1}{2}(\sqrt{2} + 1)log2$

$S = 1.1633$

$[S] = 1$

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@Vraj Mehta Nice question! $\heartsuit \heartsuit \heartsuit \heartsuit { ❤ }{ ❤ }{ ❤ }{ ❤ }$