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 Tetra-series Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 10/31/2009, 02:40 PM (This post was last modified: 10/31/2009, 02:46 PM by Gottfried.) If I Euler-sum the list of coefficients, which I get with the formula at the superroot-msg for heights h=0..63 , then the resulting powerseries seems to begin with Code:`0 + 1/2x  -1/2 x^2 + 1/4 x^3  -1/6 x^4 + 5/12 x^5 + ??? + ??? , ... ]`where for the question-marks I needed higher Euler-orders. Because this is tidy to avoid the Euler-summation at all we can do a trick: Let S(x,h) be the formal powerseries for the height h for S(x,h) = (1+x)^(1+x)^...^(1+x) - 1 and S(x) the series for the limit when h-> infinity, then by definition, AS(x) = S(x,0) - S(x,1) + S(x,2) - ... + // Euler-sum Since the coefficients of any height converge to that of the S(x)-series I compute the difference D(x,h) = S(x,h) - S(x) and rewrite AS(x) = D(x,0) - D(x,1) + D(x,2) ... + aeta(0)*S(x) where aeta(0) is the alternating zeta-series zeta(0) meaning aeta(0) = 1-1+1-1+1-... = 1/2 Because the coefficients with index k

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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