# Sum of Reciprocals of Sixth Powers Alternating in Sign

## Theorem

 $\ds \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^6}$ $=$ $\ds \frac 1 {1^6} - \frac 1 {2^6} + \frac 1 {3^6} - \frac 1 {4^6} + \cdots$ $\ds$ $=$ $\ds \frac {31 \pi^6} {30 \, 240}$

## Proof

 $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^6}$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6} - \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^6}$ separating odd positive terms from even negative terms $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^6} - \frac 1 {64} \sum_{n \mathop = 1}^\infty \frac 1 {n^6}$ $\ds$ $=$ $\ds \frac {\pi^6} {960} - \frac 1 {64} \sum_{n \mathop = 1}^\infty \frac 1 {n^6}$ Sum of Reciprocals of Sixth Powers of Odd Integers $\ds$ $=$ $\ds \frac {\pi^6} {960} - \frac 1 {64} \times \frac {\pi^6} {945}$ Riemann Zeta Function of 6 $\ds$ $=$ $\ds \frac {\pi^6 \paren {63 - 1} } {60 \, 480}$ $\ds$ $=$ $\ds \frac {31 \pi^6} {30 \, 240}$

$\blacksquare$