Question

If $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\ldots .=\frac{\pi}{4}$, Then value of $\frac{1}{1 \times 3}+\frac{1}{5 \times 7}+\frac{1}{9 \times 11}+\ldots$ is

• A. $\pi / 8$
• B. $\pi / 6$
• C. $\pi / 4$
• D. $\pi / 36$

Answer 1

Answer: (A)

We have,
$ \frac{\pi}{4}=\left(1-\frac{1}{3}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{9}-\frac{1}{11}\right)+\ldots . $
$= \frac{2}{1 \times 3}+\frac{2}{5 \times 7}+\frac{2}{9 \times 1}+\ldots .$
$\Rightarrow \frac{1}{1 \times 3}+\frac{1}{5 \times 7}+\frac{1}{9 \times 11}+\ldots=\frac{\pi}{8}$