# Sum of Reciprocals of Sixth Powers of Odd Integers

## Theorem

 $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6}$ $=$ $\ds 1 + \dfrac 1 {3^6} + \dfrac 1 {5^6} + \dfrac 1 {7^6} + \dfrac 1 {9^6} + \cdots$ $\ds$ $=$ $\ds \dfrac {\pi^6} {960}$

## Proof

 $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^6}$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^6} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6}$ $\ds$ $=$ $\ds \frac 1 {64} \sum_{n \mathop = 1}^\infty \frac 1 {n^6} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6}$ $\ds \leadsto \ \$ $\ds \dfrac {\pi^6} {945}$ $=$ $\ds \frac 1 {64} \times \dfrac {\pi^6} {945} + \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6}$ Riemann Zeta Function of 6 $\ds \leadsto \ \$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6}$ $=$ $\ds \dfrac {\pi^6} {945} - \frac 1 {64} \times \dfrac {\pi^6} {945}$ $\ds$ $=$ $\ds \dfrac {\paren {64 - 1} \pi^6} {945 \times 64}$ $\ds$ $=$ $\ds \dfrac {\pi^6} {960}$

$\blacksquare$

## Also presented as

This result can also be seen presented as:

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^6} = \dfrac {\pi^6} {960}$