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

Posted 9 years ago
 One of the growing known formulas for the MRB constant, here B, is 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]; {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 as x grew large. So I entered the following. Limit[etaMM[x, 30]/etaMM[x + 1, 30], x -> Infinity] -2.730717679880512180842720 N[-3/Log, 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)^2, 30] 7.45681904721200739937942566098  I then looked to see if -3/Log)^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)^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}]] 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] Table[etaMM[1 + x, 30], {x, 1, 10}] // TableForm 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)^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:= Limit[etaMM[x, 30 + 2]/etaMM[x + 2, 30 + 2], x -> Infinity] - N[(-3/Log)^2, 30] Out= 0.*10^-28 In:= Limit[etaMM[x, 30]/etaMM[x + 3, 30], x -> Infinity] - N[(-3/Log)^3, 30] Out= 0.*10^-26 In:= Limit[etaMM[x, 30]/etaMM[x + 4, 30], x -> Infinity] - N[(-3/Log)^4, 30] Out= 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:= Table[ Limit[etaMM[x, 30 + u]/etaMM[x + u, 30 + u], x -> Infinity] - N[(-3/Log)^u, 30], {u, 1, 10}] Out= {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}