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infinite alternating series of increasing tower-height/ reloaded
#1
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)<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
Reply
#2
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)<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).

The according plot

   

Gottfried
Gottfried Helms, Kassel
Reply
#3
Wow, nice graph! looks similar to the (x^y == y^x) function (what I call the exponential commutator (or exponential reversal) function).

I think the name you use is far too long.

If I might suggest a name, I would say alternating series of hyper-powers, since without qualifier "series" could also mean "infinite series", and "hyper-powers" or more specifically hyper-4-powers (spow), is what you describe as "increasing tower-height".

Andrew Robbins
Reply
#4
According to GFR's terms this would be the alternating series of towers.

Sorry, I just finished reading his post.

Andrew Robbins
Reply
#5
andydude Wrote:If I might suggest a name, I would say alternating series of hyper-powers, since without qualifier "series" could also mean "infinite series", and "hyper-powers" or more specifically hyper-4-powers (spow), is what you describe as "increasing tower-height".

Andrew Robbins

Well, I'm not sure, what the best naming scheme is. The number of variants for series grows.

If we have {b,x}^^h ,then we have

1) same base, increasing height (or level, GFR) of towers, top exponent x
{b,x}^^0 - {b,x}^^1 + {b,x}^^2 - ...

2) same base, same height, increasing top exponent
{b,0}^^h - {b,1}^^h + {b,2}^^h - ...

3) increasing base
{1,x}^^h - {2,x}^^h + {3,x}^^h - ...

Version 1) suggests the most naive idea, to possibly extend the characteristics of a geometric series. But I didn't find a remarkable relation yet.

Version 2) embeds the geometric series as a special case and generalizes the sum of geometric series of positive and negative exponents. After I found the relation, which I presented already here, I tend to use a name related to geometric series for this version

Version 3) would extend the harmonic series, or zeta series. In fact, I'm already fiddling with that series, but with no obviously interesting result so far.

Generally I feel, that the use of "hyper" should be reserved for attributing the *property* of being of higher order. So this is not an absolute qualifier, but always relative - relative to some other, which may be common in a context, but this context changes with the evolution of knowledge. So "tetra" may be the best choice. (The use of "hyper-geometric" was a similar bad choice, but may be understandable. At least, the related base, to what it is hyper, is preserved in the name here).

Before I'll stick to a final naming convention, I'd suggest to better see, in which context these operations / series are occuring in nature. We know, that the tetration-operation of height/level 2 occurs in the description of high-energy heat-equations and of ecological systems. If, for instance, version 1) would express some meaningful thing here, I'd bet, the most usable, significant and memorizable name comes from it.

Names like tetra-geometric series (for version 2) and tetra-harmonic-series (for version 3) would be my favourites, where for tetra-height of 1 the leading "tetra" could then simply be omitted.

For the version 1) I have thus no idea except the full description. "alternating series of tetra towers of increasing height" would express the precise meaning... "alternating series of tetra-growth towers" or "alternating tetra-growth-series"... but, well, the last was awful....
One may possibly allude to the picture of such a tower as a "stairway to heaven" ;-), finally...

Gottfried
Gottfried Helms, Kassel
Reply


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