Bernoulli Number Calculator (Compute Bₙ)

What are Bernoulli numbers?

Bernoulli numbers are a sequence of rational numbers that appear throughout number theory, combinatorics, and analysis. They are usually written as B₀, B₁, B₂, … and can be defined via the generating function

Formula: x / (e^x − 1) = ∑ n = 0 B^n x^n / (n !)

$\frac{x}{e^x-1}=\sum _{n=0}{B}^n\frac{x^n}{n!}$

In words, the coefficients in the power series expansion of x / (e^x − 1) are the Bernoulli numbers. The first few are

• B₀ = 1
• B₁ = −1/2
• B₂ = 1/6
• B₃ = 0
• B₄ = −1/30

A striking pattern is that all odd-index Bernoulli numbers beyond B₁ are zero: B₃ = B₅ = B₇ = … = 0. Non-zero even-index values grow quickly in magnitude and alternate in sign.

Historical background

Bernoulli numbers are named after Jacob Bernoulli (1655–1705), who studied them while working on formulas for sums of integer powers. For example, to compute

Formula: ∑ k = 1 n k^2

$\sum _{k=1}^n{k}^2$

Bernoulli discovered that such sums can be expressed using polynomials in n whose coefficients are built from Bernoulli numbers. Leonhard Euler later extended these ideas and revealed deep links between Bernoulli numbers, special functions, and series expansions.

Introduction: How the Bernoulli Number Calculator works

This calculator computes the Bernoulli number B_n for a given nonnegative integer n using the Akiyama–Tanigawa algorithm. The method constructs a triangular array of rational numbers and reads off B_n from the last entry in the n‑th row.

Conceptually, the algorithm proceeds as follows:

1. Choose a nonnegative integer n. The algorithm will compute all Bernoulli numbers up to B_n.
2. Initialize an array A of length n + 1.
3. For each m from 0 to n, set A[m] = 1 / (m + 1).
4. For each m from 1 to n, update A backwards: for k from m down to 1, replace A[k − 1] with k × (A[k − 1] − A[k]).
5. After the loop, A[0] contains B_m. In particular, after the final step, A[0] is B_n.

This procedure is numerically efficient for moderate n and well suited to an in-browser implementation.

Key formulas involving Bernoulli numbers

Bernoulli numbers appear in many fundamental formulas. Some of the most important include:

• Values of the Riemann zeta function at negative integers:

  Formula: ζ(− n) = − B^n+1 / (n + 1)

  $\zeta (-n)=-\frac{B^{n+1}}{n+1}$

  for n ≥ 1. This ties Bernoulli numbers directly to analytic number theory.

• Euler–Maclaurin summation formula, which links sums and integrals and uses Bernoulli numbers in the correction terms.
• Closed forms for sums of powers, such as

  Formula: ∑ k = 1 n k^p

  $\sum _{k=1}^n{k}^p$

  where the coefficients of the resulting polynomial in n involve Bernoulli numbers up to B_p.

How to use: Using the Bernoulli Number Calculator

The form above is designed to be simple:

1. Enter a nonnegative integer n in the input field.
2. Submit the form to compute B_n.
3. The calculator returns the Bernoulli number B_n. Implementations often provide both a rational representation (numerator and denominator) and, optionally, a decimal approximation.

In many contexts you only need a few initial values, for example B₀ through B₁₀. However, the algorithm can handle considerably larger n before numerical precision or performance become issues.

Interpreting the results

When you compute B_n, keep the following patterns in mind:

• B₀ = 1 and B₁ = −1/2 are special base values.
• For all odd n ≥ 3, B_n = 0.
• Non-zero even-index Bernoulli numbers alternate in sign and grow rapidly in magnitude.
• The sign of B_2m for m ≥ 1 follows a regular pattern that shows up in many asymptotic expansions.

If your result is zero for an odd n ≥ 3, that is expected and reflects a deep symmetry of the generating function. If your result has a large numerator and denominator, that is also normal: Bernoulli numbers quickly become complicated rationals.

Worked example

Suppose you want to compute B₄ using the calculator.

1. Set n = 4 in the input field.
2. The calculator runs the Akiyama–Tanigawa algorithm internally and returns B₄ = −1/30.

You can verify this value using the generating function. Expanding x / (e^x − 1) as a power series gives

Formula: x / (e^x − 1) = 1 − x / 2 + x^2 / 12 − x^4 / 720 + ⋯

$\frac{x}{e^x-1}=1-\frac{x}{2}+\frac{x^2}{12}-\frac{x^4}{720}+\cdots$

Comparing this with the defining expansion

Formula: ∑ n = 0 ∞ B^n x^n / (n !)

$\sum _{n=0}^{\infty }{B}^n\frac{x^n}{n!}$

shows that B₄/4! = −1/720, so B₄ = −1/30, matching the calculator output.

Sample values and parity pattern

The table below lists several initial Bernoulli numbers and highlights the vanishing of odd indices beyond B₁.

Only even indices (plus n = 1) yield non-zero Bernoulli numbers. This pattern is closely related to the symmetry of the generating function x / (e^x − 1) around the origin.

Applications of Bernoulli numbers

Bernoulli numbers are used in many areas:

• Summing powers of integers: closed-form expressions for sums like ∑_k=1ⁿ k^p involve Bernoulli numbers up to B_p.
• Approximating sums with integrals: the Euler–Maclaurin formula uses Bernoulli numbers to express discretization error, which is essential in numerical analysis.
• Special functions: derivatives and expansions of trigonometric and hyperbolic functions often display Bernoulli numbers in their coefficients.
• Analytic number theory: relations like ζ(−n) = −B_n+1 / (n + 1) directly connect Bernoulli numbers to the Riemann zeta function.

Limitations and assumptions of this calculator

While the Akiyama–Tanigawa algorithm is efficient, any practical implementation must work within numeric and performance limits. Typical assumptions and constraints include:

• Input must be a nonnegative integer. Fractional or negative indices are not allowed, because Bernoulli numbers are defined for integer n.
• Maximum index. For very large n, intermediate values can grow extremely large. Implementations usually enforce an upper bound (for example, a few hundred or a few thousand) to avoid long computation times or memory issues.
• Representation of results. Internally, rational arithmetic or high-precision integers may be used. Displayed values may be exact fractions, decimal approximations, or both. For very large n, decimal output may be rounded.
• Rounding and precision. If decimals are shown to a fixed number of places, small rounding errors are possible in the displayed approximation, even if the internal rational value is exact.
• Performance expectations. For moderate indices (for example, n up to a few hundred), the algorithm should run quickly in a browser. For much larger n, you should expect longer runtimes or potential failure due to resource limits.

For research-grade or very high-index computations, dedicated computer algebra systems or arbitrary-precision libraries are recommended.

Visualising Bernoulli numbers

Although this page focuses on numerical computation rather than plotting, you can export values and create your own charts in external tools. If you plot B_n against n, you will see an alternating pattern of positive and negative spikes at even indices and zeros at most odd indices. If you instead plot |B_n| on a logarithmic scale, you will notice that the points lie close to a straight line, indicating an almost exponential growth in magnitude.

These visual patterns help explain why only a few Bernoulli numbers significantly contribute to many practical formulas: higher-order terms become very large but are also multiplied by high powers or factorials, so their net effect is controlled.