First slide

Special Series in Sequences and Series

Question

If $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \dots = \frac{π}{4},$ then value of $\frac{1}{1 \cdot 3} + \frac{1}{5 \cdot 7} + \frac{1}{9 \cdot 11} + \dots$ is

Moderate

A

$\frac{π}{8}$
B

$\frac{π}{6}$
C

$\frac{π}{4}$
D

$\frac{π}{34}$

Solution

We have
$\begin{array}{l} \frac{π}{4} = (1 - \frac{1}{3}) + (\frac{1}{5} - \frac{1}{7}) + (\frac{1}{9} - \frac{1}{11}) + \dots \\ = \frac{2}{1 \cdot 3} + \frac{2}{5 \cdot 7} + \frac{2}{9 \cdot 11} + \dots \\ \Rightarrow \frac{1}{1 \cdot 3} + \frac{1}{5 \cdot 7} + \frac{1}{9 \cdot 11} + \dots = \frac{π}{8} \end{array}$

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