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Bernoulli Number — Definition, Formula & Examples

Bernoulli numbers are a specific sequence of rational numbers that appear naturally in formulas for sums of integer powers, in the Taylor series of certain functions, and throughout number theory.

The Bernoulli numbers BnB_n are the sequence of rational numbers defined by the exponential generating function tet−1=∑n=0∞Bntnn!\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}, or equivalently by the recursion ∑k=0n(n+1k)Bk=0\sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 for n≥1n \geq 1, with B0=1B_0 = 1.

Key Formula

Bn=−1n+1∑k=0n−1(n+1k)Bk,n≥1B_n = -\frac{1}{n+1}\sum_{k=0}^{n-1} \binom{n+1}{k} B_k, \quad n \geq 1
Where:
  • BnB_n = The $n$th Bernoulli number
  • B0B_0 = Initial value, equal to 1
  • (n+1k)\binom{n+1}{k} = Binomial coefficient

How It Works

You can compute Bernoulli numbers one at a time using the recursion. Start with B0=1B_0 = 1, then for each n≥1n \geq 1, solve ∑k=0n(n+1k)Bk=0\sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 for BnB_n. This gives Bn=−1n+1∑k=0n−1(n+1k)BkB_n = -\frac{1}{n+1}\sum_{k=0}^{n-1} \binom{n+1}{k} B_k. All odd-indexed Bernoulli numbers beyond B1B_1 are zero, so in practice you only need to track even indices.

Worked Example

Problem: Compute the Bernoulli numbers B₀, B₁, and B₂ using the recursive formula.
B₀: By definition, the first Bernoulli number is
B0=1B_0 = 1
B₁: For n = 1, solve the equation involving the sum with k = 0 to 1:
B1=−12∑k=00(20)B0=−12(1)(1)=−12B_1 = -\frac{1}{2}\sum_{k=0}^{0}\binom{2}{0}B_0 = -\frac{1}{2}(1)(1) = -\frac{1}{2}
B₂: For n = 2, sum over k = 0 and k = 1:
B2=−13[(30)B0+(31)B1]=−13[1+3(−12)]=−13(−12)=16B_2 = -\frac{1}{3}\left[\binom{3}{0}B_0 + \binom{3}{1}B_1\right] = -\frac{1}{3}\left[1 + 3\left(-\tfrac{1}{2}\right)\right] = -\frac{1}{3}\left(-\tfrac{1}{2}\right) = \frac{1}{6}
Answer: B0=1B_0 = 1, B1=−12B_1 = -\tfrac{1}{2}, B2=16B_2 = \tfrac{1}{6}. The sequence continues B3=0B_3 = 0, B4=−130B_4 = -\tfrac{1}{30}, and so on.

Why It Matters

Bernoulli numbers give closed-form expressions for power sums like 1k+2k+⋯+nk1^k + 2^k + \cdots + n^k via Faulhaber's formulas. They also appear in the Euler–Maclaurin summation formula and connect to the Riemann zeta function at negative integers, making them central to analytic number theory.

Common Mistakes

Mistake: Confusing the two sign conventions for B₁. Some textbooks define B₁ = +1/2 instead of −1/2.
Correction: Check which convention your course uses. The generating function t/(et−1)t/(e^t - 1) gives B1=−1/2B_1 = -1/2, while the alternative t/(et−1)+t/2t/(e^t - 1) + t/2 shifts B1B_1 to +1/2+1/2. Both are standard.