Bernoulli Number Calculator (Compute Bₙ)

What are Bernoulli numbers?

Bernoulli numbers are rational coefficients that recur in number theory, combinatorics, and analysis. Written B₀, B₁, B₂, …, they are defined by the generating function

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

Thus, Bernoulli numbers are obtained from the coefficients in the power-series expansion of x / (e^x − 1). The first few are

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

A defining pattern of this sequence is that every odd-index Bernoulli number after B₁ vanishes: B₃ = B₅ = B₇ = … = 0. The nonzero even-index values alternate in sign and rapidly increase in magnitude.

Historical background of Bernoulli numbers

Bernoulli numbers take their name from Jacob Bernoulli (1655–1705), who encountered them while deriving formulas for sums of integer powers. For example, one such sum is

Formula: ∑ k = 1 n k^2

Bernoulli found that power sums can be written as polynomials in n with coefficients involving these numbers. Leonhard Euler later developed related connections to special functions and infinite series.

Introduction: How the Bernoulli Number Calculator works

This Bernoulli Number Calculator evaluates B_n for the nonnegative integer index n with the Akiyama–Tanigawa recurrence. It builds a triangular sequence of floating-point values and takes B_n from the first array entry after the n-th pass.

Conceptually, the Bernoulli recurrence used on this page proceeds as follows:

1. Choose a nonnegative integer n. The recurrence processes rows from 0 through 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 each pass, A[0] is the Bernoulli value for that pass; after the final pass it is B_n.

The calculator also directly returns zero for odd indices greater than 1, which is the established parity property of Bernoulli numbers and avoids unnecessary floating-point work.

Key formulas involving Bernoulli numbers

Bernoulli numbers supply coefficients in several central formulas, including the following:

• Values of the Riemann zeta function at negative integers:

  Formula: ζ(− n) = − B^n+1 / (n + 1)

  for n ≥ 1. This identity connects the sequence to analytic number theory.

• Euler–Maclaurin summation formula, whose correction terms use Bernoulli numbers to relate a finite sum to an integral.
• Closed forms for sums of powers, such as

  Formula: ∑ k = 1 n k^p

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

How to use: Using the Bernoulli Number Calculator

To calculate a particular Bernoulli number on this page, use the index field as follows:

1. Enter a nonnegative integer n in the input field.
2. Submit the form to compute B_n.
3. Read the displayed decimal value and its positive, negative, or zero label. The result can be copied with the Copy Result button.

The calculator displays six decimal places for ordinary-sized results and exponential notation for values with magnitude at least one million. This makes it useful for checking initial values and signs, while exact fractional forms should be obtained from a symbolic or arbitrary-precision tool when they are required.

Interpreting the results of a Bernoulli number calculation

When this calculator reports B_n, the index n provides important context for reading the value:

• B₀ = 1 and B₁ = −1/2 are exceptional starting values.
• For every odd n ≥ 3, B_n = 0.
• Nonzero even-index Bernoulli numbers alternate in sign and grow rapidly in magnitude.
• The sign of B_2m for m ≥ 1 follows the alternating pattern used in many asymptotic expansions.

A zero result at an odd index n ≥ 3 is therefore expected, not a missing calculation. At larger even indices, a decimal output may be very large or use exponential notation; that presentation reflects the growth of the Bernoulli number rather than a change in the definition.

Worked example: calculating the Bernoulli number B₄

To find B₄ with the Bernoulli Number Calculator, enter 4 as the index n.

1. Set n = 4 in the input field.
2. The calculator applies the recurrence and displays B₄ as approximately −0.033333, with a negative sign.

The value can be checked from the Bernoulli generating function. Its power-series expansion begins

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

Comparing this with the defining expansion

Formula: ∑ n = 0 ∞ B^n x^n / (n !)

shows that B₄/4! = −1/720, so B₄ = −1/30. The calculator’s six-place decimal is the displayed approximation of that fraction.

Sample Bernoulli values and parity pattern

This table gives early Bernoulli values and makes the zero pattern at odd indices after 1 explicit.

Only even indices, together with n = 1, produce nonzero Bernoulli numbers. This behavior follows from the symmetry properties of x / (e^x − 1) near the origin.

Applications of Bernoulli numbers

Bernoulli numbers matter whenever discrete sums, power series, or special-function expansions need systematic correction coefficients:

• 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 corrections in numerical analysis.
• Special functions: derivatives and expansions of trigonometric and hyperbolic functions often have Bernoulli numbers as coefficients.
• Analytic number theory: relations such as ζ(−n) = −B_n+1 / (n + 1) directly connect Bernoulli numbers to the Riemann zeta function.

Limitations and assumptions of this Bernoulli Number Calculator

This browser calculator evaluates Bernoulli numbers with JavaScript floating-point arithmetic, so its output is intended as a numerical value rather than an exact rational certificate. Its practical constraints are:

• Input must be a nonnegative integer. Fractional or negative indices are not accepted by this calculator.
• Maximum index. The page limits n to 300 to avoid floating-point overflow in the browser.
• Displayed representation. Results are shown as decimals, with exponential notation for sufficiently large magnitudes; numerator-and-denominator fractions are not displayed.
• Floating-point precision. Rounding can affect displayed digits, especially as the recurrence’s intermediate values become large.
• Parity shortcut. For odd indices greater than 1, the page returns the mathematically exact value zero directly.

For exact high-index Bernoulli fractions, use software that implements arbitrary-precision rational arithmetic and compare its convention for B₁ with the generating function shown above.

Visualising Bernoulli numbers

Bernoulli values calculated here can be copied into another program if you want to graph the sequence by index. A plot of B_n against n has zeros at odd indices beyond 1 and alternating positive and negative values at even indices.

Plotting the absolute values of the nonzero even-index terms on a logarithmic scale makes their rapid growth easier to see. In applications such as asymptotic expansions, those large coefficients are paired with other terms, so the size of B_n alone does not determine the size of a final correction.