• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Tetra-series Gottfried Ultimate Fellow Posts: 765 Threads: 119 Joined: Aug 2007 06/29/2008, 09:41 PM (This post was last modified: 06/30/2008, 06:21 AM by Gottfried.) A curious result from study of the tetra-series. (text updated) I considered the "reverse" of the tetra-series problem. Instead of asking for the a_lternating s_um of powertowers of increasing p_ositive heights (asp) Code:`asp(x,dxp) = dxp°0(x) - dxp°1(x) + dxp°2(x) - ... + ...` where dxp(x) = exp(x)-1 and dxp°h(x) is the h'th integer-iterate. I asked for a function tf(x) where Code:`asp(x,tf) = (e^x-1)/2 = tf°0(x) - tf°1(x) + tf°2(x) - ... +...`so I ask: can (e^x - 1)/2 be represented by a tetra-series of a function tf(x) and what would that function look like? Using the matrix-operator-approach I got the result Code:```tf(x) = x - x^2 + 2/3*x^3 - 3/4*x^4 + 11/15*x^5 - 59/72*x^6 + 379/420*x^7 - 331/320*x^8        + 1805/1512*x^9 - 282379/201600*x^10 + 3307019/1995840*x^11 - 6152789/3110400*x^12        + 616774003/259459200*x^13 - 3212381993/1117670400*x^14 + 54372093481/15567552000*x^15        - 594671543783/139485265920*x^16 + 58070127447587/11115232128000*x^17        - 1209735800444267/188305108992000*x^18 + 26776614379573099/3379030566912000*x^19        - 209181772596680209/21341245685760000*x^20 + 1034961114326994557/85151570286182400*x^21        - 80852235077445729119/5352384417988608000*x^22        + 2210690796475549862239/117509166994931712000*x^23        - 18624665294361841906483/793412278431252480000*x^24        + 379264261780067802109819/12926008369442488320000*x^25        - 6584114267874407529534167/179240649389602504704000*x^26        + 5046681464320089079803469/109576837040799744000000*x^27        - 326480035696597942691643978259/5646080455772478898176000000*x^28        + 327920863401689931801359966641/4511103058030460180889600000*x^29        - 418419411682443365665393881223739/4573325169175707907522560000000*x^30        + 15798888070625404329026746075454779/137047310902965380295426048000000*x^31        + O(x^32)```which can be determined to arbitrary many coefficients by a recursive process on rational numbers. The float-representation is Code:```tf(x) = 1.00000000000*x - 1.00000000000*x^2 + 0.666666666667*x^3 - 0.750000000000*x^4       + 0.733333333333*x^5 - 0.819444444444*x^6 + 0.902380952381*x^7 - 1.03437500000*x^8       + 1.19378306878*x^9 - 1.40068948413*x^10 + 1.65695596841*x^11 - 1.97813432356*x^12       + 2.37715218038*x^13 - 2.87417649515*x^14 + 3.49265533084*x^15 - 4.26332874559*x^16       + 5.22437379435*x^17 - 6.42433870711*x^18 + 7.92434807834*x^19 - 9.80176020073*x^20       + 12.1543397362*x^21 - 15.1058348510*x^22 + 18.8129220299*x^23 - 23.4741329327*x^24       + 29.3411740841*x^25 - 36.7333765543*x^26 + 46.0560972611*x^27 - 57.8241911808*x^28       + 72.6919467774*x^29 - 91.4912883306*x^30 + 115.280540468*x^31 + O(x^32)``` so I assume, that this series tf(x) has radius of convergence limited to about |x|<0.7 Moreover, the iterates of this function seem always to be of a similar form, so the alternating sum of the found coefficients of the iterated functions at like powers of x is divergent for each coefficient (but may be Euler-summed). So this result must be considered in more detail next, since I had inconsistency of the matrix-method with serial summation either for increasing positive or for increasing negative heights. However, for x=1/2 or x=1/3 or smaller we can accelerate convergence of asp() by Euler-summation such that I get good (?) approximation to the six'th digit for x=1/3 using the truncated series with 31 terms only. The process for the generation of these coefficients is a bit tedious yet; so I don't have -for instance - the function, whose iterations must be non-alternating summed to get the exp(x)-1 value (or (exp(x)-1)/2 or some other scalar multiple) which -as I guess- could have better range of convergence. I'll post the result, if I got it. Gottfried Gottfried Helms, Kassel « 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

 Possibly Related Threads... Thread Author Replies Views Last Post Perhaps a new series for log^0.5(x) Gottfried 3 675 03/21/2020, 08:28 AM Last Post: Daniel Taylor series of i[x] Xorter 12 13,258 02/20/2018, 09:55 PM Last Post: Xorter Taylor series of cheta Xorter 13 14,327 08/28/2016, 08:52 PM Last Post: sheldonison Derivative of E tetra x Forehead 7 9,999 12/25/2015, 03:59 AM Last Post: andydude [integral] How to integrate a fourier series ? tommy1729 1 2,777 05/04/2014, 03:19 PM Last Post: tommy1729 Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 17,776 02/04/2012, 05:02 AM Last Post: Kouznetsov Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 3,067 09/04/2011, 05:59 AM Last Post: Gottfried What is the convergence radius of this power series? JmsNxn 9 16,815 07/04/2011, 09:08 PM Last Post: JmsNxn An alternate power series representation for ln(x) JmsNxn 7 14,445 05/09/2011, 01:02 AM Last Post: JmsNxn weird series expansion tommy1729 2 4,619 07/05/2010, 07:59 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)