${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+{\sin }^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+\mathrm{......}+{\sin }^{-1}\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)+\mathrm{....}\infty =$

$T_n={\sin }^{-1}\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n\left(n+1\right)}}={\mathrm{Tan}}^{-1}\frac{\sqrt{n}-\sqrt{n-1}}{1+\sqrt{n}\sqrt{n-1}}$

     $={\mathrm{Tan}}^{-1}\sqrt{n}-{\mathrm{Tan}}^{-1}\sqrt{n-1}$

$\therefore$$S_{\infty }=\left\{{\mathrm{Tan}}^{-1}1-{\mathrm{Tan}}^{-1}0\right\}+\{{\mathrm{Tan}}^{-1}\sqrt{2}-{\mathrm{Tan}}^{-1}1\}+\mathrm{......}+{\mathrm{Tan}}^{-1}\infty$

$={\mathrm{Tan}}^{-1}\infty -{\mathrm{Tan}}^{-1}0=\frac{\pi }{2}$