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Help further pattern in the infinite limit of nth eta derivatives of n

One of the growing known formulas for the MRB constant, here B, is enter image description here

where you use consecutive mth Dirichlet eta derivatives of m. (I hope I said that right.)

Below etaMM[m, pr] is a module that rapidly computes the above mentioned Dirichlet eta derivatives of m for m>1 to "pr" decimals.

Clear[n]; 
etaMM[m_, pr_] := 
 Module[{a, d, s, k, b, c}, a[j_] := Log[j + 1]^m/(j + 1)^m;
  n = Floor[1.32 pr];
  d = Cos[n ArcCos[3]];
  {b, c, s} = {-1, -d, 0};
  Do[c = b - c;
   s = s + c a[k];
   b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}];
  Return[N[s/d, pr] (-1)^m]];

When experimenting with etaMM it seemed that the limit etaMM(x)/etaMM(x+1) as x goes to infinity converged to -3/Log[3] as x grew large. So I entered the following.

Limit[etaMM[x, 30]/etaMM[x + 1, 30], x -> Infinity]

-2.730717679880512180842720

N[-3/Log[3], 30]

-2.73071767988051218084272049721

I then put my attention on the similar limit of etaMM(x)/etaMM(x+2) and entered,

Limit[etaMM[x, 30]/etaMM[x + 2, 30], x -> Infinity]

7.4568190472120073993794257

N[(-3/Log[3])^2, 30]

7.45681904721200739937942566098

I then looked to see if -3/Log[3])^n fit the pattern for the above mentioned limit of etaMM(x)/etaMM(x+n) by entering the following command.

Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - 
  N[(-3/Log[3])^n, 30], {n, 1, 10}]

{-1.37235287940800822169041524488*10^40, 
 2.79444910535423129879510807704*10^41, \
-7.63085157751707944108437643731*10^41, 
 2.08377013152699855184232233095*10^42, \
-4.24307018055104941251691654111*10^43, 
 1.15866267590045475709082282171*10^44, \
-3.16398065409903564894573215540*10^44, 
 6.44264535810872405469659678725*10^45, \
-1.75930455845876064949571024220*10^46, \
-1.31188157413621969593288955467*10^47}

Please join in the conversation if you have anything to add! In particular I would like to find a shortcut for finding etaMM(x+n) from etaMM(x). This may be hard to do because etaMM(x) is not at all well behaved for small x and works its way to near 0 for larger x.; see the following.

ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}]]

enter image description here

It is my guess that etaMM is a decaying, semi-oscillating function:

ListPlot[Table[etaMM[1 + x, 30], {x, 1, 10}], 
 PlotRange -> {{1, 10}, {-0.005, 0.005}}, Joined -> True]

enter image description here

Table[etaMM[1 + x, 30], {x, 1, 10}] // TableForm

enter image description here

POSTED BY: Marvin Ray Burns

As some of you already figured out, something is awry with my use and interpretation of

Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - 
  N[(-3/Log[3])^n, 30], {n, 1, 10}]

{-1.37235287940800822169041524488*10^40, 
 2.79444910535423129879510807704*10^41, \
-7.63085157751707944108437643731*10^41, 
 2.08377013152699855184232233095*10^42, \
-4.24307018055104941251691654111*10^43, 
 1.15866267590045475709082282171*10^44, \
-3.16398065409903564894573215540*10^44, 
 6.44264535810872405469659678725*10^45, \
-1.75930455845876064949571024220*10^46, \
-1.31188157413621969593288955467*10^47}

because, to the contrary, I can enter and get the following.

In[13]:= Limit[etaMM[x, 30 + 2]/etaMM[x + 2, 30 + 2], x -> Infinity] -
  N[(-3/Log[3])^2, 30]

Out[13]= 0.*10^-28

In[11]:= Limit[etaMM[x, 30]/etaMM[x + 3, 30], x -> Infinity] - 
 N[(-3/Log[3])^3, 30]

Out[11]= 0.*10^-26

In[16]:= Limit[etaMM[x, 30]/etaMM[x + 4, 30], x -> Infinity] - 
 N[(-3/Log[3])^4, 30]

Out[16]= 0.*10^-25

It looks like I've got n doing double work; it's global in etaMM and the table both. This works, though:

In[58]:= Table[
 Limit[etaMM[x, 30 + u]/etaMM[x + u, 30 + u], x -> Infinity] - 
  N[(-3/Log[3])^u, 30], {u, 1, 10}]

Out[58]= {0.*10^-26, 0.*10^-28, 0.*10^-29, 0.*10^-29, 0.*10^-28, 
 0.*10^-28, 0.*10^-27, 0.*10^-27, 0.*10^-27, 0.*10^-26}
POSTED BY: Marvin Ray Burns
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