# Sum of Reciprocals of Fourth Powers Alternating in Sign

## Theorem

 $\ds \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^4}$ $=$ $\ds \frac 1 {1^4} - \frac 1 {2^4} + \frac 1 {3^4} - \frac 1 {4^4} + \cdots$ $\ds$ $=$ $\ds \dfrac {7 \pi^4} {720}$

## Proof

 $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^4}$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4} - \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^4}$ separating odd positive terms from even negative terms $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n - 1}^4} - \frac 1 {16} \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ $\ds$ $=$ $\ds \frac {\pi^4} {96} - \frac 1 {16} \sum_{n \mathop = 1}^\infty \frac 1 {n^4}$ Sum of Reciprocals of Fourth Powers of Odd Integers $\ds$ $=$ $\ds \frac {\pi^4} {96} - \frac 1 {16} \times \frac {\pi^4} {90}$ Riemann Zeta Function of 4 $\ds$ $=$ $\ds \frac {\pi^4 \paren {15 - 1} } {1440}$ $\ds$ $=$ $\ds \frac {7 \pi^4} {720}$

$\blacksquare$