proof of Faulhaber’s formula
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 that gives us the appropriate sum of powers on the other side. Equating coefficients of powers of then gives the result.
But this is also a geometric series, so
Equating coefficients of we get
which proves the first equality.
If is a polynomial, write for the coefficient of in . Then
and thus if , iterating, we get
Then using the fact that , we have
Now reverse the order of summation (i.e. replace by ) to get
|Title||proof of Faulhaber’s formula|
|Date of creation||2013-03-22 18:43:50|
|Last modified on||2013-03-22 18:43:50|
|Last modified by||rm50 (10146)|