If frac{1}{1^4} + frac{1}{2^4} + frac{1}{3^4} | vedclass

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(A) Let $S = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}$.We can split the sum into odd and even terms: $S = \sum_{n=1,3,5,\dots}^{\infty} \f

Vedclass · Accepted Answer

$\frac{\pi^4}{96}$

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(A) Let $S = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}$.We can split the sum into odd and even terms: $S = \sum_{n=1,3,5,\dots}^{\infty} \frac{1}{n^4} + \sum_{n=2,4,6,\dots}^{\infty} \frac{1}{n^4}$.Let $S_{odd} = \sum_{n=1,3,5,\dots}^{\infty} \frac{1}{n^4}$.The even terms can be written as $\sum_{k=1}^{\infty} \frac{1}{(2k)^4} = \frac{1}{2^4} \sum_{k=1}^{\infty} \frac{1}{k^4} = \frac{1}{16} S$.Thus,$S = S_{odd} + \frac{1}{16} S$.$S_{odd} = S - \frac{1}{16} S = \frac{15}{16} S$.Substituting $S = \frac{\pi^4}{90}$,we get $S_{odd} = \frac{15}{16} 	imes \frac{\pi^4}{90} = \frac{\pi^4}{16 	imes 6} = \frac{\pi^4}{96}$.

If $\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \dots + \infty = \frac{\pi^4}{90}$,then the value of $\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \dots + \infty$ is

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