Not sure if this will help anything, but here's a graph showing the root test for the first 33 Taylor coefficients in b for \( ^{1/2} b \) expanded about b = 1.42. I used the regular formula
\( ^{t} b = \lim_{n \rightarrow \infty} \log_b^n\left(F - \left(F - \exp^n_b(1)\right) \log(F)^t\right) \)
(this is just a rearrangement of the formula mentioned in my original post to this thread)
with n = 8192, 2048 digits of precision (just to be safe) and the simple, straightforward numerical differentiation via the difference quotient (delta step 10^-20). Took about an hour(!) to compute due to the extreme precision... I could probably do some more if I could parallelize the job but Pari/GP doesn't have parallel computation facilities (no threads, no MPI, no nothing like that). I suppose I could recode the formula as a C or Fortran program for a parallel job, interfaced with an arbitrary-precision package like GMP or MPFR (I think Pari/GP uses something like this under the hood), and crunch even more coeffs, because crunching these coeffs is tediously slow, esp. if I have to add more precision (I'm pretty sure up to the 30th or 31st coeff is right, not sure about the accuracy of the last 2 but they should be, though only a recalc with more precision will tell.).
Coeffs:
Graph of root test (testing for convergence radius, as you mention):
The big question, of course, is does it dip below 0.42, the expected radius of convergence, and stay below it, out to infinity? If it does, then that would indicate a closer singularity. If it drops to 0.02466... then that would indicate a singularity at \( b = e^{1/e} \). (Of course, the graph alone isn't a proof.)
\( ^{t} b = \lim_{n \rightarrow \infty} \log_b^n\left(F - \left(F - \exp^n_b(1)\right) \log(F)^t\right) \)
(this is just a rearrangement of the formula mentioned in my original post to this thread)
with n = 8192, 2048 digits of precision (just to be safe) and the simple, straightforward numerical differentiation via the difference quotient (delta step 10^-20). Took about an hour(!) to compute due to the extreme precision... I could probably do some more if I could parallelize the job but Pari/GP doesn't have parallel computation facilities (no threads, no MPI, no nothing like that). I suppose I could recode the formula as a C or Fortran program for a parallel job, interfaced with an arbitrary-precision package like GMP or MPFR (I think Pari/GP uses something like this under the hood), and crunch even more coeffs, because crunching these coeffs is tediously slow, esp. if I have to add more precision (I'm pretty sure up to the 30th or 31st coeff is right, not sure about the accuracy of the last 2 but they should be, though only a recalc with more precision will tell.).
Coeffs:
Code:
a_0 = 1.246220033102832033637391168
a_1 = 0.4479211001478450057502788291
a_2 = -0.1944285662380565757140341278
a_3 = 0.1431678738611080328964523214
a_4 = -0.1449673997741212873243507213
a_5 = 0.1821593012238922403824879947
a_6 = -0.2634074264259034150890239199
a_7 = 0.4170984777615359795375604795
a_8 = -0.7022041478879047165932451401
a_9 = 1.235350783631857820858433698
a_10 = -2.247058971388908630541707138
a_11 = 4.196782696009382012696224987
a_12 = -8.009130240912803756451562486
a_13 = 15.56210572774662623695298001
a_14 = -30.70293022611311616071677783
a_15 = 61.37465434532134439666317750
a_16 = -124.0937577676786262630256434
a_17 = 253.4276217341761588591503345
a_18 = -522.1524296306182164337119282
a_19 = 1084.317735417423446029022288
a_20 = -2267.611476993890273952854904
a_21 = 4772.099072723291344681868206
a_22 = -10098.44557704069780712228148
a_23 = 21465.93884981489863800142138
a_24 = -45708.11493010257551050715076
a_25 = 96243.61297567708695783556454
a_26 = -185475.2857026224205407755534
a_27 = 186483.8146778817735118583707
a_28 = -451252.6193153016119357011782
a_29 = 73309968.03777933438145964027
a_30 = -1983477241.135794800710911594
a_31 = 19226263007.10400347804764423
a_32 = 346411096888.4104287944798334Graph of root test (testing for convergence radius, as you mention):
The big question, of course, is does it dip below 0.42, the expected radius of convergence, and stay below it, out to infinity? If it does, then that would indicate a closer singularity. If it drops to 0.02466... then that would indicate a singularity at \( b = e^{1/e} \). (Of course, the graph alone isn't a proof.)
