numbers come from the coefficients in the
Taylor expansion of x/(ex-1).
They can be defined recursively by setting
B0=1, and then using
The first few Bernoulli numbers are B0=1, B1=-1/2, B2=1/6, B3=0, B4=-1/30, B5=0, B6=1/42, B7=0, B8=-1/30, B9=0, and B10=5/66. Notice that all of the odd terms, B2n+1 (n > 1), are zero; and the even terms alternate in sign.
These numbers can also be defined using the Riemann zeta function as follows
Finally, using Stirling's formula, we have
The Bernoulli numbers first appeared in the posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli. Euler used them to express the sums of equal powers of consecutive integers. They also are important in classical assaults of Fermat's Last Theorem.
See Also: Regular
Related pages (outside of this work)
- A Bibliography of Bernoulli Numbers (excellent!)
- Ribenboim95 (pp. 217-218)
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]