First slide

Properties of ITF

Question

If $\sum_{r = 1}^{\infty} \tan^{- 1} \frac{1}{2 r^{2}} = t,$ then tan $t$ is equal to

Moderate

A

0
B

1
C

$\sqrt{3}$
D

$1 / \sqrt{3 .}$

Solution

$\begin{array}{l} \sum_{r = 1}^{\infty} \tan^{- 1} \frac{1}{2 r^{2}} = \sum_{1}^{\infty} [\tan^{- 1} (2 r + 1) - \tan^{- 1} (2 r - 1)] \\ = \tan^{- 1} 3 - \tan^{- 1} 1 + \tan^{- 1} 5 - \tan^{- 1} 3 + \dots \\ = π / 2 - π / 4 = π / 4 \end{array}$

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