Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
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.
Reply


Messages In This Thread
RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 08:24 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Perhaps a new series for log^0.5(x) Gottfried 3 430 03/21/2020, 08:28 AM
Last Post: Daniel
Question Taylor series of i[x] Xorter 12 12,680 02/20/2018, 09:55 PM
Last Post: Xorter
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,637 01/17/2017, 05:10 AM
Last Post: JmsNxn
  Taylor series of cheta Xorter 13 13,769 08/28/2016, 08:52 PM
Last Post: sheldonison
  2015 Continuum sum conjecture tommy1729 3 3,584 05/26/2015, 12:24 PM
Last Post: tommy1729
  Another way to continuum sum! JmsNxn 6 6,823 06/06/2014, 05:09 PM
Last Post: MphLee
  [integral] How to integrate a fourier series ? tommy1729 1 2,678 05/04/2014, 03:19 PM
Last Post: tommy1729
  Continuum sum = Continuum product tommy1729 1 2,786 08/22/2013, 04:01 PM
Last Post: JmsNxn
  applying continuum sum to interpolate any sequence. JmsNxn 1 2,998 08/18/2013, 08:55 PM
Last Post: tommy1729
  Powerful way to perform continuum sum JmsNxn 7 8,247 08/12/2013, 07:17 PM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)