Riemann Zeta Function at Even Integers/Corollary
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Corollary to Riemann Zeta Function at Even Integers
\(\ds B_{2 n}\) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \paren {1 + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \cdots}\) |
where:
- $B_n$ are the Bernoulli numbers
- $n$ is a positive integer.
Proof
\(\ds \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}\) | \(=\) | \(\ds \map \zeta {2 n}\) | Riemann Zeta Function at Even Integers | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }\) | Definition of Riemann Zeta Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^{n + 1} B_{2 n}\) | \(=\) | \(\ds \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }\) | multiplying both sides by $\dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} }$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^{2 n + 2} B_{2 n}\) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }\) | multiplying both sides by $\paren {-1}^{n + 1}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds B_{2 n}\) | \(=\) | \(\ds \paren {-1}^{n + 1} \dfrac {\paren {2 n}!} {2^{2 n - 1} \pi^{2 n} } \sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} }\) | $\paren {-1}^{2 n + 2} = 1$ as $2 n + 2$ is even |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 21$: Series involving Bernoulli and Euler Numbers: $21.8$