DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-27

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

Definition

Write $e^{t} = \exp (t)$ for the complex exponential and set $h (t) = \sum_{m = 0}^{\infty} t^{m} / (m + 1)!$ . The exponential series converges everywhere, $e^{t} - 1 = t h (t)$ , and $h (0) = 1$ . Hence $1 / h (t)$ is holomorphic on some neighbourhood of $0$ by A convergent complex power series with nonzero constant term has a convergent reciprocal power series locally. It equals $t / (e^{t} - 1)$ wherever $t \neq 0$ in that neighbourhood; at $0$ , the quotient means this removable extension. Its Maclaurin expansion has the 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. Since $h (t) = 1 + t / 2 + \dots$ , the reciprocal coefficient formula gives $B_{0} = 1$ and $B_{1} = - 1 / 2$ .

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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)