It is given that $\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + ... + to\ \infty = \frac{\pi^{4}}{90}$. Then, $\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + ...\infty$ is equal to

Question

Dr. Abubacker · Accepted Answer

(a)
Since, $\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + ...\infty = \frac{\pi^{4}}{90}$
$\Rightarrow \frac{\pi}{90} = \left( \frac{1}{1^{4}} + \frac{1}{3^{4}} + ...\infty \right) + \left( \frac{1}{2^{4}} + \frac{1}{4^{4}} + \frac{1}{6^{4}} + ...\infty \right)$
$\Rightarrow \frac{\pi^{4}}{90} = \left( \frac{1}{1^{4}} + \frac{1}{3^{4}} + ...\infty \right) + \frac{1}{2^{4}}\left( \frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + ...\infty \right)$
$\Rightarrow \frac{\pi^{4}}{90} = \left( \frac{1}{1^{4}} + \frac{1}{3^{4}} + ...\infty \right) + \frac{1}{16}\left( \frac{\pi^{4}}{90} \right)$
$\Rightarrow \frac{\pi^{4}}{96} = \frac{1}{1^{4}} + \frac{1}{3^{4}} + ...\infty$

Q. It is given that $\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + ... + to\ \infty = \frac{\pi^{4}}{90}$. Then, $\frac{1}{1^{4}} + \frac{1}{3^{4}} + \frac{1}{5^{4}} + ...\infty$ is equal to

Explanation

Reviewed by Dr. Abubacker

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