Bernoulli Numbers Calculator

Compute Bernoulli numbers, which are a sequence of rational numbers important in number theory and have applications in calculus, Fourier analysis, and elsewhere.

Input Parameters

Enter a non-negative integer. Higher values may take longer to compute.

Calculation Results

Result

B₁ = -1/2

Decimal form: -0.5

Calculation Formula

Bn can be computed using the Akiyama–Tanigawa algorithm:

E0(n) = [1, 1], Ek(n) = [1] + [Σj=0k-1 Ek-1(n)(n+j)/(k+j)] for k ≥ 1

Bn = En(n)₁n+1

Bernoulli Numbers Calculator Usage Guide

Learn how to use the Bernoulli Numbers Calculator and understand their significance in mathematics.

What are Bernoulli Numbers?

Bernoulli numbers Bn are a sequence of rational numbers which are very important in number theory. They appear in the expansions of trigonometric functions, in the coefficients of the Taylor series of the tangent and secant functions, in the Euler–Mascheroni constant, and in many other areas of mathematics.

How to Use This Calculator

  1. Enter a non-negative integer value for the index n in the "Index (n)" field.
  2. Click the "Calculate" button to compute the Bernoulli number Bn.
  3. The calculator will display the exact fraction form of Bn and its decimal approximation.

Significance of Bernoulli Numbers

Bernoulli numbers have numerous applications:

  • They appear in the formula for the sum of the natural numbers:1 + 2 + 3 + ... + n = n(n+1)/2 + B1n2/2 + B2n3/3 + ...

  • They are used in the expansion of tan(x) and sec(x) functions.
  • They appear in the Euler–Mascheroni constant γ formula: γ = ∑n=1∞ (-1)n+1B2n / (2n(2n-1)).
  • They are related to the Riemann zeta function ζ(s) at odd integers s = 2n + 1.

Limitations

For very large values of n (n > 100), the computation may become slow or the numbers may become extremely large, depending on the implementation and browser capabilities.