${\sin }^{-1}\frac{1}{\sqrt{2}}+{\sin }^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+{\sin }^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+{\sin }^{-1}\frac{2-\sqrt{3}}{\sqrt{20}}+{\sin }^{-1}\frac{\sqrt{5}-2}{\sqrt{30}}+\mathrm{....}\infty =$

$=\sum _1^{\infty }{\sin }^{-1}\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}$

$=\sum _1^{\infty }{\sin }^{-1}[\frac{1}{\sqrt{n}}\sqrt{\frac{n}{n+1}}-\frac{1}{\sqrt{n+1}}\sqrt{\frac{n-1}{n}}]$

$=\sum _1^{\infty }{\sin }^{-1}[\frac{1}{\sqrt{n}}\sqrt{1-\frac{1}{n+1}}-\frac{1}{\sqrt{n+1}}\sqrt{1-\frac{1}{n}}]$

$=\underset{n\to \infty }{Lt}\sum _1^n[{\sin }^{-1}\frac{1}{\sqrt{n}}-{\sin }^{-1}\frac{1}{\sqrt{n+1}}]$

$=\underset{n\to \infty }{Lt}[\frac{\pi }{2}-{\sin }^{-1}\frac{1}{\sqrt{n+1}}]=\frac{\pi }{2} (by telescopic)$