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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

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

Vedclass · Answer

(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

Similar Questions

The sum of the series $3 + 33 + 333 + \dots + n$ terms is

Find the sum of infinite terms of the series $1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\cdots$

The sequence $\frac{5}{\sqrt{7}}, \frac{6}{\sqrt{7}}, \sqrt{7}, \dots$ is

For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,
Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.
Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.

If the sum of $1 + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + \dots$ to $n$ terms is $S$,then $S$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module