Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Rational sums of inverse powers of fixed points of e
#12
Yes, they converge very rapidly. The fixed points increase by approximately 2*pi*I as we move away from the real line, so they form an arithmetic sequence to a first approximation. Therefore, they'll converge about as fast as the partial sums of the zeta function. So yes, we shouldn't need very many fixed points to calculate the coefficients of the rational numbers, though I haven't bothered to figure out if some minimum number of fixed points would suffice for all sums.

My main interest in calculating 100,000 fixed points was to generate continued fractions for each sum, and try to use those to provide strong numerical evidence that the infinite sums indeed converge on rational numbers.
~ Jay Daniel Fox
Reply


Messages In This Thread
RE: Rational sums of inverse powers of fixed points of e - by jaydfox - 11/21/2007, 06:20 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  tetration from alternative fixed point sheldonison 22 30,521 12/24/2019, 06:26 AM
Last Post: Daniel
  Thoughts on hyper-operations of rational but non-integer orders? VSO 2 773 09/09/2019, 10:38 PM
Last Post: tommy1729
  Inverse Iteration Xorter 3 2,759 02/05/2019, 09:58 AM
Last Post: MrFrety
  Inverse super-composition Xorter 11 13,773 05/26/2018, 12:00 AM
Last Post: Xorter
  Are tetrations fixed points analytic? JmsNxn 2 3,168 12/14/2016, 08:50 PM
Last Post: JmsNxn
  the inverse ackerman functions JmsNxn 3 6,148 09/18/2016, 11:02 AM
Last Post: Xorter
  Rational operators (a {t} b); a,b > e solved JmsNxn 30 41,420 09/02/2016, 02:11 AM
Last Post: tommy1729
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,637 03/19/2016, 10:44 AM
Last Post: fivexthethird
  Inverse power tower functions tommy1729 0 1,957 01/04/2016, 12:03 PM
Last Post: tommy1729
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 10,915 01/01/2016, 03:58 PM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)