the odd Bernoulli numbers are zero

Recall that, for k0, the Bernoulli numbersMathworldPlanetmathPlanetmath Bk are defined as the coefficients in the Taylor expansionMathworldPlanetmath:

tet-1=k0Bktkk!. (1)

Just to name a few:


If k3 is odd then Bk=0.


From the right hand side of (1) we extract the term corresponding to k=1:

tet-1=-t2+k0,k1Bktkk!. (2)


tet-1+t2=k0,k1Bktkk! (3)

and the left hand side can be rewritten as:

tet-1+t2=2t+t(et-1)2(et-1)=t2et+1et-1=t2et/2+e-t/2et/2-e-t/2. (4)

Hence, if one replaces t by -t then (4) is unchanged. Since (4) is the left hand side of (3), the quantity


is also unchanged when t is exchanged by -t, and so we must have Bk=(-1)kBk for k1. We conclude that if k3 and k is odd, Bk=0. ∎

Title the odd Bernoulli numbers are zero
Canonical name TheOddBernoulliNumbersAreZero
Date of creation 2013-03-22 15:12:04
Last modified on 2013-03-22 15:12:04
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 11B68
Related topic KummersCongruence
Related topic CongruenceOfClausenAndVonStaudt