Riemann Zeta Function at Non-Positive Integers/Examples/Zeta(-1)
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Example of Use of Riemann Zeta Function at Non-Positive Integers
- $\map \zeta {-1 } = -\dfrac 1 {12} $
Proof
Follows directly from the Riemann Zeta Function at Non-Positive Integers:
Explicit derivation illustrated below:
\(\ds \map \zeta {-1 }\) | \(=\) | \(\ds \paren {-1}^n \frac {B_{n + 1} } {n + 1}\) | Riemann Zeta Function at Non-Positive Integers | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^1 \frac {B_{1 + 1} } {1 + 1}\) | Entering $n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {B_{2} } {2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {12}\) | From Definition:Bernoulli Numbers/Sequence, $B_2 = \dfrac 1 6 $ |
$\blacksquare$