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 Tetra-series andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/03/2009, 05:04 AM (This post was last modified: 11/03/2009, 05:07 AM by andydude.) If F is all zeroes except for a one somewhere, then that represents an $x^n$ function. In general for integer n, the G's look like this $ \begin{tabular}{rl} \frac{1}{x^4} &= 5 - 14x + 35({}^{2}x) - \frac{245}{3}({}^{3}x) + \frac{1957}{12}({}^{4}x) + \cdots \\ \frac{1}{x^3} &= 4 - 9x + 19({}^{2}x) - 39({}^{3}x) - \frac{817}{12}({}^{4}x) + \cdots \\ \frac{1}{x^2} &= 3 - 5x + \frac{17}{2}({}^{2}x) - 15({}^{3}x) - \frac{533}{24}({}^{4}x) + \cdots \\ \frac{1}{x} &= 2 - 2x + \frac{5}{2}({}^{2}x) - \frac{11}{3}({}^{3}x) + \frac{35}{8}({}^{4}x) + \cdots \\ 1 &= 1({}^{0}x) + 0 \\ x &= 0 + 1({}^{1}x) \\ x^2 &= -1 + x + \frac{3}{2}({}^{2}x) - \frac{2}{3}({}^{3}x) - \frac{5}{24}({}^{4}x) + \cdots \\ x^3 &= -2 + \frac{7}{2}({}^{2}x) - \frac{41}{24}({}^{4}x) + \frac{37}{20}({}^{5}x) + \cdots \\ x^4 &= -3 - 2x + 5({}^{2}x) + 3({}^{3}x) - \frac{37}{12}({}^{4}x) + \cdots \end{tabular}$ The first coefficient seems to have a pattern in it, but this is just because $g_0 = f(1) - f'(1) = 1 - n$. Oh, and another weird thing: ${}^{\infty}x = 0 + {}^{n}x$ when approximated in this way. « Next Oldest | Next Newest »

 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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