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Evaluate the value of the infinite sum \(\sin^{-1}\!\left(\frac{1}{\sqrt{2}}\right)+\sin^{-1}\!\left(\frac{\sqrt{2}-1}{\sqrt{6}}\right)+\sin^{-1}\!\left(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}\right)+\cdots+\sin^{-1}\!\left(\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)+\cdots\).

1. \(\pi/8\)
2. \(\pi/4\)
3. \(\pi/2\)
4. \(\pi\)

Correct answer: \(\pi/2\)

Solution

The infinite sum converges to \\frac{C0}{2} because each term in the series represents the inverse sine of a value that approaches the limit of 1 as n increases, leading to the cumulative angle approaching \\frac{C0}{2}.

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