This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BernoulliB

 BernoulliB[n]gives the Bernoulli number . BernoulliBgives the Bernoulli polynomial .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• The Bernoulli polynomials satisfy the generating function relation .
• The Bernoulli numbers are given by .
• BernoulliB can be evaluated to arbitrary numerical precision.
First 10 Bernoulli numbers:
Bernoulli polynomials:
First 10 Bernoulli numbers:
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Bernoulli polynomials:
 Out[1]=
 Scope   (3)
Plot Bernoulli polynomials:
 Applications   (6)
Find sums of powers using BernoulliB:
Compare with direct summation:
Set up an Euler-Maclaurin integration formula:
Use it for :
Compare with the exact summation result:
Plot roots of Bernoulli polynomials in the complex plane:
Show the approach of Bernoulli numbers to a limiting form:
The denominator of Bernoulli numbers is given by the von Staudt-Clausen formula:
Compute Bernoulli numbers in modular arithmetic modulo a prime:
Find BernoulliB numbers from their generating function:
Find BernoulliB polynomials from their generating function:
Algorithmically produced results are frequently expressed using Zeta instead of BernoulliB:
When entered in the traditional form, is not automatically interpreted as a Bernoulli number:
Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:
The 20000 Bernoulli number can be computed in under a second: