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:
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.
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)<s<e^(1/e) are bounded in their absolute values and thus can conventionally be summed by, for instance, Euler-or Cesaro summation. (I added those results for comparision to the table).
But since for s>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
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)<s<e^(1/e) are bounded in their absolute values and thus can conventionally be summed by, for instance, Euler-or Cesaro summation. (I added those results for comparision to the table).
But since for s>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
