Question

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

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

Answer 1

Answer: (A)

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