# Sum of Reciprocals of Fourth Powers of Odd Integers

## Theorem

 $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4}$ $=$ $\ds 1 + \dfrac 1 {3^4} + \dfrac 1 {5^4} + \dfrac 1 {7^4} + \dfrac 1 {9^4} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^4} {96}$

## Proof

 $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^4} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4}$ $\ds$ $=$ $\ds \frac 1 {16} \sum_{n \mathop = 1}^\infty \frac 1 {n^4} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4}$ $\ds \leadsto \ \$ $\ds \dfrac {\pi^4} {90}$ $=$ $\ds \frac 1 {16} \times \dfrac {\pi^4} {90} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4}$ Riemann Zeta Function of 4 $\ds \leadsto \ \$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4}$ $=$ $\ds \dfrac {\pi^4} {90} - \frac 1 {16} \times \dfrac {\pi^4} {90}$ $\ds$ $=$ $\ds \dfrac {\paren {16 - 1} \pi^4} {90 \times 16}$ $\ds$ $=$ $\ds \dfrac {\pi^4} {96}$

$\blacksquare$

## Also presented as

This result can also be seen presented as:

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^4} = \dfrac {\pi^4} {96}$