First slide

Special Series in Sequences and Series

Question

If $\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots$ upto $\infty = \frac{π^{2}}{6}$ , then value of $\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \dots$ up to $\infty$ is

Moderate

A

$\frac{π^{2}}{4}$
B

$\frac{π^{2}}{6}$
C

$\frac{π^{2}}{8}$
D

$\frac{π^{2}}{12}$

Solution

We have $\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \dots$ upto $\infty$
$= \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \frac{1}{5^{2}} + \frac{1}{6^{2}} \dots$ upto $\infty$
$\begin{array}{l} - \frac{1}{2^{2}} [1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots] \\ = \frac{π^{2}}{6} - \frac{1}{4} (\frac{π^{2}}{6}) = \frac{π^{2}}{8} \end{array}$

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