First slide

Properties of ITF

Question

The sum of the series $\cot^{- 1} 2 + \cot^{- 1} 8 + \cot^{- 1} 18 + \cot^{- 1} 32 + \dots$ is

Moderate

A

$\frac{π}{2}$
B

$\frac{π}{4}$
C

$\frac{π}{6}$
D

none of these

Solution

The given series
$= \cot^{- 1} 2 (1)^{2} + \cot^{2} 2 (2)^{2} + \cot^{- 1} 2 (3)^{2} + \cot^{- 1} 2 (4)^{2} + \dots$
$\begin{matrix} ∴ T_{n} = \cot^{- 1} (2 n^{2}) = \tan^{- 1} \frac{1}{2 n^{2}} \\ = \tan^{- 1} [\frac{(2 n + 1) - (2 n - 1)}{1 + (2 n + 1) (2 n - 1)}] \\ = \tan^{- 1} (2 n + 1) - \tan^{- 1} (2 n - 1) \\ ∴ T_{1} = \tan^{- 1} 3 - \tan^{- 1} 1 \end{matrix}$
$\begin{aligned} T_{2} = \tan^{- 1} 5 - \tan^{- 1} 3 \\ T_{3} = \tan^{- 1} 7 - \tan^{- 1} 5 \end{aligned}$
and so on.
$\begin{array}{l} ∴ S_{n} = \tan^{- 1} (2 n + 1) - \tan^{- 1} 1 \\ ∴ lim_{n \to \infty} S_{n} = \tan^{- 1} \infty - \frac{π}{4} = \frac{π}{2} - \frac{π}{4} = \frac{π}{4} \end{array}$

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