proof of Faulhaber’s formula

Theorem 0.1.

If kN,2nZ, then


where the Bi are the Bernoulli numbersMathworldPlanetmathPlanetmath and bi the Bernoulli polynomialsMathworldPlanetmathPlanetmath.

The exponential generating function for the Bernoulli numbers is


We develop an equation involving sums of Bernoulli numbers on one side, and a simple generating involving powers of e that gives us the appropriate sum of powers on the other side. Equating coefficients of powers of x then gives the result.

To get a generating function where the coefficient of xn/n! is m=1n-1mk, we can use

m=0n-1emx =m=0n-1k=0mkxkk!

But this is also a geometric seriesMathworldPlanetmath, so

k=0n-1ekx =1-enx1-ex

Equating coefficients of xk/k! we get

m=1n-1mk =i=0k1k-i+1(ki)Bink+1-i

which proves the first equality.

If f(x) is a polynomial, write [xr]f(x) for the coefficient of xr in f(x). Then


and thus if rk, iterating, we get


Then using the fact that bk=kbk-1, we have

1nbk(x) =1k+1(bk+1(n)-bk+1(1))=1k+1r=0k+1[xr]bk+1(x)(nr-1)

Now reverse the order of summation (i.e. replace r by k+1-r) to get

Title proof of Faulhaber’s formula
Canonical name ProofOfFaulhabersFormula
Date of creation 2013-03-22 18:43:50
Last modified on 2013-03-22 18:43:50
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 11B68