Question

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

• A. $\frac{\pi }{8}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{36}$

Answer 1

Answer: (A)

$\frac{\pi }{8}$

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