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
Question Taylor series of i[x] Xorter 12 10,073 02/20/2018, 09:55 PM
Last Post: Xorter
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,298 01/17/2017, 05:10 AM
Last Post: JmsNxn
  Taylor series of cheta Xorter 13 10,746 08/28/2016, 08:52 PM
Last Post: sheldonison
  2015 Continuum sum conjecture tommy1729 3 3,025 05/26/2015, 12:24 PM
Last Post: tommy1729
  Another way to continuum sum! JmsNxn 6 5,947 06/06/2014, 05:09 PM
Last Post: MphLee
  [integral] How to integrate a fourier series ? tommy1729 1 2,268 05/04/2014, 03:19 PM
Last Post: tommy1729
  Continuum sum = Continuum product tommy1729 1 2,389 08/22/2013, 04:01 PM
Last Post: JmsNxn
  applying continuum sum to interpolate any sequence. JmsNxn 1 2,621 08/18/2013, 08:55 PM
Last Post: tommy1729
  Powerful way to perform continuum sum JmsNxn 7 7,307 08/12/2013, 07:17 PM
Last Post: JmsNxn
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 15,562 02/04/2012, 05:02 AM
Last Post: Kouznetsov



Users browsing this thread: 1 Guest(s)