Riemann Zeta Function of 4/Proof 4
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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
| \(\ds \map \zeta 4\) | \(=\) | \(\ds \paren {-1}^3 \dfrac {B_4 2^3 \pi^4} {4!}\) | Riemann Zeta Function at Even Integers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^3 \paren {-\dfrac 1 {30} } \dfrac {2^3 \pi^4} {4!}\) | Definition of Sequence of Bernoulli Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\dfrac 1 {30} } \paren {\dfrac 8 {24} } \pi^4\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\pi^4} {90}\) | simplifying |
$\blacksquare$