Question

The sum of the series $\frac{1}{1.3}+\frac{1}{5.7}+\frac{1}{9.11}+...to\infty$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $1$
• D. $\frac{\pi }{8}$

Answer 1

The correct option is D $\frac{\pi }{8}$

Given series: $\Theta =\tan \Theta -\frac{1}{3}{\tan }^3\Theta +\frac{1}{5}{\tan }^5\Theta -\frac{1}{7}{\tan }^7\Theta +\frac{1}{9}{\tan }^9\Theta -\frac{1}{11}{\tan }^{11}\Theta +...\infty$
Substitute $\Theta =\frac{\pi }{4}$
$\frac{\pi }{4}=(1-\frac{1}{3})+(\frac{1}{5}-\frac{1}{7})+(\frac{1}{9}-\frac{1}{11})+...\infty$
$\Rightarrow \frac{2}{1.3}+\frac{2}{5.7}+\frac{2}{9.11}+...\infty =\frac{\pi }{4}$
Therefore, $\frac{1}{1.3}+\frac{1}{5.7}+\frac{1}{9.11}+...\infty =\frac{\pi }{8}$
Ans: D