12/15/2009, 06:56 AM
Hmm. I think I may have figured out what's going on. It seems like Faulhaber's formula fails to sum
\( \sum_{n=0}^{x-1} e^{qn} \)
for what appears to be \( q \ge 2\pi \). Below that it works and sums it to, apparently \( \frac{e^{qx} - 1}{e^q - 1} \). So that would provide an analytic continuation to higher q-values, thus the formula given is valid in a sense, but direct application of the Faulhaber formula will not work. It seems that a combination of analytic continuation and divergent summation theory applied to Faulhaber and transseries theory is needed to put together a comprehensive theory of continuum sums of general analytic functions.
I find the appearance of \( 2\pi \) interesting, considering it is the magnitude of the imaginary period of the exponential function. I don't know if that is significant or not.
\( \sum_{n=0}^{x-1} e^{qn} \)
for what appears to be \( q \ge 2\pi \). Below that it works and sums it to, apparently \( \frac{e^{qx} - 1}{e^q - 1} \). So that would provide an analytic continuation to higher q-values, thus the formula given is valid in a sense, but direct application of the Faulhaber formula will not work. It seems that a combination of analytic continuation and divergent summation theory applied to Faulhaber and transseries theory is needed to put together a comprehensive theory of continuum sums of general analytic functions.
I find the appearance of \( 2\pi \) interesting, considering it is the magnitude of the imaginary period of the exponential function. I don't know if that is significant or not.
