Riemann Zeta Function of 6/Proof 3
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Theorem
The Riemann zeta function of $6$ is given by:
| \(\ds \map \zeta 6\) | \(=\) | \(\ds \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\pi^6} {945}\) | ||||||||||||
| \(\ds \) | \(\approx\) | \(\ds 1 \cdotp 01734 \, 3 \ldots\) |
Proof
| \(\ds \sum_{n \mathop = 1}^{\infty} \frac 1 {n^6}\) | \(=\) | \(\ds \map \zeta 6\) | Definition of Riemann Zeta Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1}^4 \frac {B_6 2^5 \pi^6} {6!}\) | Riemann Zeta Function at Even Integers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {42} \cdot \frac {2^5 \pi^6} {6!}\) | Definition of Sequence of Bernoulli Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {32 \pi^6} {42 \cdot 720}\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\pi^6} {945}\) |
$\blacksquare$