# ProofWiki:Sandbox/Template

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## Theorem

The Riemann zeta function of $4$ is given by:

\(\displaystyle \map \zeta 4\) | \(=\) | \(\displaystyle \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\pi^4} {90}\) | |||||||||||

\(\displaystyle \) | \(\approx\) | \(\displaystyle 1 \cdotp 08232 \, 3 \ldots\) |

## Proof

\(\displaystyle \map \zeta 4\) | \(=\) | \(\displaystyle \paren{\map \zeta 2 }^2 - 2 \dfrac { \pi^4} {5!}\) | Squaring Zeta of 2 produces Zeta of 4 plus two times the sum associated with the 4th power term in the sin(x)/x expansion | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac { \pi^4} {36} - \dfrac { \pi^4} {60}\) | simplifying | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {\pi^4} {90}\) | simplifying |

$\blacksquare$