Find the sum of the infinite series S= ^ -1 ( 1 2 )+ ^ -1 ( 2 -1 6 )+ ^ -1 ( 3 - 2 2…

MATHEMATICS · Subjective · CBSE · Mathematics · Inverse Trigonometric Functions · Inverse Trigonometric Functions

Problem

Find the sum of the infinite series S=sin⁡−1(12)+sin⁡−1(2−16)+sin⁡−1(3−223)+⋯+sin⁡−1(n−n−1n(n+1))+⋯ .S=\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)+\sin^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+\sin^{-1}\left(\frac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}\right)+\cdots+\sin^{-1}\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)+\cdots.

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