---
tag: 0RAS24E1R5
kind: definition
name: Bernoulli number
msc: [11B68]
defines: 0KC00K7KY4
sources:
  - url: https://en.wikipedia.org/wiki/Bernoulli_number
    locator: Generating function
  - url: https://en.wikipedia.org/wiki/Bernoulli_number
    locator: Recursive definition
---

## Definition

The Bernoulli numbers $B_0, B_1, B_2, \dots$ are the rational numbers given by the power series expansion
$$\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}.$$
Equivalently, $B_0 = 1$ and $\sum_{k=0}^{n} \binom{n+1}{k} B_k = 0$ for each $n \ge 1$.

## Remarks

The first values are $B_0 = 1$, $B_1 = -\frac{1}{2}$, $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$ and $B_6 = \frac{1}{42}$. Also $B_n = 0$ for each odd $n \ge 3$.
Some authors use the convention $B_1 = +\frac{1}{2}$, from the series $\frac{t e^t}{e^t - 1}$.
They appear in the formulas for the sums of powers $1^k + 2^k + \dots + n^k$, and in the values $\zeta(2n)$ of the Riemann zeta function.
They are named after Jacob Bernoulli. Seki Takakazu tabulated them in his Katsuyō Sanpō of 1712.

## Explanations

### L1 human

The Bernoulli numbers are a sequence of fractions that appear in many formulas, for example in the formula for $1^k + 2^k + \dots + n^k$.
You can compute them one after another from a simple recurrence.