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 want to find the sum of the odd terms: $S_{odd} = \sum_{n=0}^{\infty} \frac{1}{(

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

Vedclass · Answer

(A) Let $S = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}$.We want to find the sum of the odd terms: $S_{odd} = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^4} = \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \dots$.We know that $S = S_{odd} + S_{even}$,where $S_{even} = \sum_{n=1}^{\infty} \frac{1}{(2n)^4}$.$S_{even} = \frac{1}{2^4} \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{1}{16} S$.Therefore,$S_{odd} = S - S_{even} = S - \frac{1}{16} S = \frac{15}{16} S$.Substituting the value of $S = \frac{\pi^4}{90}$:$S_{odd} = \frac{15}{16} 	imes \frac{\pi^4}{90} = \frac{1}{16} 	imes \frac{\pi^4}{6} = \frac{\pi^4}{96}$.

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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