First slide

Properties of ITF

Question

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

Difficult

A

$π$
B

$\frac{π}{2}$
C

$\frac{π}{4}$
D

$\frac{3 π}{2}$

Solution

$T_{n} = \sin^{- 1} \frac{\sqrt{n} - \sqrt{n - 1}}{\sqrt{n (n + 1)}} = {Tan}^{- 1} \frac{\sqrt{n} - \sqrt{n - 1}}{1 + \sqrt{n} \sqrt{n - 1}}$
$= {Tan}^{- 1} \sqrt{n} - {Tan}^{- 1} \sqrt{n - 1}$
$∴$ $S_{\infty} = \{{Tan}^{- 1} 1 - {Tan}^{- 1} 0\} + {{Tan}^{- 1} \sqrt{2} - {Tan}^{- 1} 1} + ...... + {Tan}^{- 1} \infty$
$= {Tan}^{- 1} \infty - {Tan}^{- 1} 0 = \frac{π}{2}$

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