In this paper we present a generalization of Faulhaber's formula to
sums of arbitrary complex powers
![$m\in\mathbb{C} $](img1.gif)
.
These summation
formulas for sums of the form
![$\sum_{k=1}^{\lfloor x\rfloor}k^{m}$](img2.gif)
and
![$\sum_{k=1}^{n}k^{m}$](img3.gif)
,
where
![$x\in\mathbb{R} ^{+}$](img4.gif)
and
![$n\in\mathbb{N} $](img5.gif)
,
are based on a series acceleration involving Stirling numbers of the
first kind. While it is well-known that the corresponding expressions
obtained from the Euler-Maclaurin summation formula diverge, our summation
formulas are all very rapidly convergent.