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Theorem
The Riemann zeta function of $4$ is given by:
\(\ds \map \zeta 4\) | \(=\) | \(\ds \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi^4} {90}\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 1 \cdotp 08232 \, 3 \ldots\) |
Proof
\(\ds \map \zeta 4\) | \(=\) | \(\ds \paren{\map \zeta 2 }^2 - 2 \dfrac { \pi^4} {5!}\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac { \pi^4} {36} - \dfrac { \pi^4} {60}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\pi^4} {90}\) | simplifying |
$\blacksquare$