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 infinite alternating series of increasing tower-height/ reloaded Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 09/16/2007, 09:33 AM (This post was last modified: 09/16/2007, 09:41 PM by Gottfried.) Here I review an infinite series of tetra-towers, which I discussed some weeks ago. When I conjectured this first, it was nearly a pure heuristic; but since the eigenvalue-approach came out to be meaningful, I think, this problem has gotten a better base for discussion. The problem is still the question about eigenvalues, where the base-parameter s exceeds the bounds for the tetra-tower of infinite height: on one hand it looks, that the continuous tetration can still be described using the proposed eigensystem-decomposition, on the other hand the complex-valued solution for t=h(s), s =t^(1/t) where s>e^(1/e) is not yet better established than as an quite interesting looking proposition. The subject is the alternating series of tetra-towers of the same base but increasing height: Code:.   AS(s) = 1 - s + s^^2 - s^^3 + s^^4 - s^^5 +...-... Clearly, for s>e^(1/e) this series diverges quickly, and for,say, s=10 we are summing here googol and googolplex already in the first 5 terms of the series: a rate of divergence which can currently not be summed by any known technique for divergent summation. Code:Example: AS(10) = 1 - 10 + 10^10 - 10^10^10 + ... - ... Since, with the matrix-method, AS(s) can be expressed using a geometric series of the matrix-operator, the operator for to obtain the values for the alternating sum can be represented as Ms = (I+Bs)^-1 and since the assumed diverging sequence of eigenvalues of Bs occur now as reciprocals this leads to a convergent sequence of eigenvalues for the oparator Ms. This consideration has a better base now and should be applicable also for the case, where the eigenvalues are complex due to non-real solutions for t, where t^(1/t)=s with s>e^(1/e). Thus this problem should be answered soon. For support by numerical computations I add results, which I got with a new computation of values for AS(s). The terms of the series are generally oscillating in sign, but for e^(-e)e(1/e) the terms grow too quickly, any conventional summation-method fails after few terms (exponent too high), a correspondent in sci.math.research even tried the "baby step-giant step" method (Shanks method) but could not obtain values for s>3. The matrix-method still provides values, in a astonishing smooth continuation of the tendency as long as using "safe" values for s. For s<5 the values seem to be precise for about 12 to 16 digits, for s>5 this gets a bit worse, but, with the applied method, for s=10.5 about 5 to 6 digits should be trustworthy. Gottfried Code:Matrix-method                   |  sumalt in Pari/Gp                                   |   V(1)~* (I+ Bs)^-1 = Y~          |   y = AS(s)   y = Y[1] = AS(s)                |     = sumalt(k=0,(-1)^k*tetra(s,k))                                   | Dim=32                            |           s        AS(s)          |    AS(s) -----------------------------------------------------------   0.500000000000  0.938253002822  | 0.938253002822                         0.600000000000  0.806376025100  | 0.806376025100                         0.700000000000  0.704392031371  | 0.704392031371                         0.800000000000  0.622421602195  | 0.622421602195                         0.900000000000  0.555271139824  | 0.555271139824                          1.00000000000  0.500000000000  | 0.500000000000                          1.10000000000  0.454762779286  | 0.454762779286                          1.20000000000  0.418151329549  | 0.418151329549                          1.30000000000  0.388800295397  | 0.388800295397                          1.40000000000  0.365258864959  | 0.365258864959                       -----------------------------------------------------------    1.50000000000  0.346148339216  |  *** for: exponent too large in exp.    1.60000000000  0.330363887727    1.70000000000  0.317099035362    1.80000000000  0.305777072512    1.90000000000  0.295982049296    2.00000000000  0.287408698053    2.10000000000  0.279828635046    2.20000000000  0.273067833614    2.30000000000  0.266991324606    2.40000000000  0.261492899921    2.50000000000  0.256487634949    2.60000000000  0.251907055589    2.70000000000  0.247694884082    2.80000000000  0.243804780370    2.90000000000  0.240198213869    3.00000000000  0.236842493055    3.10000000000  0.233709908638    3.20000000000  0.230777040831    3.30000000000  0.228023768838    3.40000000000  0.225432500367    3.50000000000  0.222987845199    3.60000000000  0.220676446560    3.70000000000  0.218486712270    3.80000000000  0.216408456685    3.90000000000  0.214432588149    4.00000000000  0.212550920480    4.10000000000  0.210756089104    4.20000000000  0.209041507295    4.30000000000  0.207401310793    4.40000000000  0.205830274317    4.50000000000  0.204323710898    4.60000000000  0.202877374284    4.70000000000  0.201487380094    4.80000000000  0.200150151524    4.90000000000  0.198862386811    5.00000000000  0.197621041067    5.10000000000  0.196423314543    5.20000000000  0.195266641263    5.30000000000  0.194148674752    5.40000000000  0.193067270079    5.50000000000  0.192020463155    5.60000000000  0.191006448967    5.70000000000  0.190023560552    5.80000000000  0.189070250087    5.90000000000  0.188145072936    6.00000000000  0.187246674933    6.10000000000  0.186373782727    6.20000000000  0.185525196731    6.30000000000  0.184699786068    6.40000000000  0.183896484897    6.50000000000  0.183114289564    6.60000000000  0.182352256131    6.70000000000  0.181609497960    6.80000000000  0.180885183168    6.90000000000  0.180178531857    7.00000000000  0.179488813094    7.10000000000  0.178815341708    7.20000000000  0.178157474956    7.30000000000  0.177514609163    7.40000000000  0.176886176405    7.50000000000  0.176271641338    7.60000000000  0.175670498208    7.70000000000  0.175082268110    7.80000000000  0.174506496511    7.90000000000  0.173942751045    8.00000000000  0.173390619597    8.10000000000  0.172849708634    8.20000000000  0.172319641794    8.30000000000  0.171800058684    8.40000000000  0.171290613867    8.50000000000  0.170790976013    8.60000000000  0.170300827180    8.70000000000  0.169819862204    8.80000000000  0.169347788165    8.90000000000  0.168884323925    9.00000000000  0.168429199704    9.10000000000  0.167982156688    9.20000000000  0.167542946657    9.30000000000  0.167111331627    9.40000000000  0.166687083493    9.50000000000  0.166269983671    9.60000000000  0.165859822749    9.70000000000  0.165456400126    9.80000000000  0.165059523649    9.90000000000  0.164669009255    10.0000000000  0.164284680610    10.1000000000  0.163906368744    10.2000000000  0.163533911701    10.3000000000  0.163167154184    10.4000000000  0.162805947215    10.5000000000  0.162450147797 for comparision, for s=10.5:                   0.162450147797  dim =32                         !        it seems there is a methodspecific minimum in this interval due to the truncation                   0.162444920277  dim= 64                   0.162445889658  dim= 80                   0.162446615009  dim= 96                   0.162447531040  dim=128                   0.162448514172  dim=256 Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 09/16/2007, 09:35 PM (This post was last modified: 09/16/2007, 10:18 PM by Gottfried.) Gottfried Wrote:For support by numerical computations I add results, which I got with a new computation of values for AS(s). The terms of the series are generally oscillating in sign, but for e^(-e)

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