DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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The Bernoulli numbers are defined by the generating series $t / (e^{t} - 1)$

Definition

Near $t = 0$ , the quotient $t / (e^{t} - 1)$ is holomorphic because $e^{t} - 1 = t + O (t^{2})$ . Its Maclaurin expansion therefore has a unique form

$\frac{t}{e^{t} - 1} = \sum_{n = 0}^{\infty} \frac{B_{n}}{n!} t^{n} .$

The coefficients $B_{n}$ are the Bernoulli numbers. This normalization gives $B_{0} = 1$ and $B_{1} = - 1 / 2$ .

Depends on

The complex exponential by its power series

Used by

Dependency tree · two levels

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Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 11 §3 (standard reference, not scraped)

The Bernoulli numbers are defined by the generating series t/(et−1)t/(et−1)