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 $

$\blacksquare$