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Mittag-Leffler series for generating continuum sum?
#3
The idea is that maybe if Faulhaber's formula does not yield a convergent formula when applied directly to a Taylor series with finite convergence radius, perhaps it would if we could apply it to a Mittag-Leffler series or some other extension of the Taylor series to a cut plane. The reasoning being that the "reason it does not converge" (if you go and read the thread "Continuum sum formula rescued?", I mention this there) may be that the Taylor series doesn't look "globally" like the function it represents due to the limited convergence (and the partial sums of it don't approach, on a "global scale", the function), while th Mittag-Leffler extension would, and thus maybe the Faulhaber formula will succeed there, when it failed on the Taylor series. But I can't give it a shot, without being able to compute that formula. My first test would be to try finding the Mittag-Leffler expansion for log(1 + z) (Taylor series at 0 is the Mercator series), and then apply Faulhaber's formula and see if we get convergence to the log-factorial.
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RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 08:24 PM

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