Dirichlet Eta Function at Non-Positive Integers
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Theorem
Let $n \ge 0$ be a integer.
Then:
- $\map \eta {-n} = \dfrac {\paren {-1}^{n + 1} \paren {2^{n + 1 } - 1} B_{n + 1} } {n + 1}$
where:
- $B_n$ is the $n$th Bernoulli number
- $\eta$ is the Dirichlet eta function
Proof
| \(\ds \map \eta s\) | \(=\) | \(\ds \paren {1 - 2^{1 - s} } \map \zeta s\) | Riemann Zeta Function in terms of Dirichlet Eta Function | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \eta {- n}\) | \(=\) | \(\ds \paren {1 - 2^{1 - \paren {-n} } } \map \zeta {-n}\) | setting $s := - n$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1 - 2^{n + 1} } \map \zeta {-n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1 - 2^{n + 1} } \paren {\paren {-1}^n \dfrac {B_{n + 1} } {n + 1} }\) | Riemann Zeta Function at Non-Positive Integers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1} \times \paren {1 - 2^{n + 1 } } \times \paren {-1} \times \paren {\paren {-1}^n \dfrac {B_{n + 1} } {n + 1} }\) | multiplying by $1$: $-1 \times -1 = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {-1}^{n + 1} \paren {2^{n + 1 } - 1 } B_{n + 1} } {n + 1}\) |
$\blacksquare$