Bernoulli number

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.