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How can I find series radius convergence using Mathematica? I know SumConvergence just give me convergence condition.

 SumConvergence[Exp[(-A*k^2 + B*k) HarmonicNumber[n]], n]

this give me:

Re[k (-B + A k)] > 1
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3
  • $\begingroup$ Also DiscreteLimit[Abs[Exp[(-A*k^2 + B*k)* HarmonicNumber[n]]]^(1/n), n -> Infinity] results in $1$, remaining the question about the convergence open. $\endgroup$
    – user64494
    Commented May 8, 2023 at 14:40
  • $\begingroup$ Do you mean the radius of convergence for its generating function? $\endgroup$
    – Greg Hurst
    Commented May 9, 2023 at 11:52
  • $\begingroup$ @ Greg Hurst, yes I want to calculate radius of convergence for this generating function $\endgroup$
    – caren
    Commented May 9, 2023 at 17:14

1 Answer 1

1
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One may apply the comparison test 1 (see Wiki for its formulation) with Mathematica 13.2 on Windows 10:

Series[Exp[(-A*k^2 + B*k) HarmonicNumber[n]], n -> Infinity] // Normal;
SumConvergence[%,n]

Re[k (-B + A k)] > 1

Next,

Reduce[Re[k (-B + A k)] > 1, k]

produces the result which takes 0.7 MB.

Addition. Its full simplification is

`(Re[A] == 0 && ((A == 0 && (B != Re[B] || 1 + Re[B] Re[k] < 0) && (B == Re[B] || 1 + Re[B] Re[k] < Im[B] Im[k])) || (A != 0 && ((B != Re[B] && Im[B] != 2 Im[A] Re[k] && Im[B] Im[k] > 1 + (2 Im[A] Im[k] + Re[B]) Re[k]) || (B == Re[B] && 1 + (2 Im[A] Im[k] + Re[B]) Re[k] < 0))) || (2 Re[k] == Im[B]/Im[A] && B != Re[B] && ((Im[2 A + B Re[B]] > 0 && Im[A] < 0) || (Im[A] > 0 && Im[2 A + B Re[B]] < 0))))) || (B != Re[B] && ((Re[A] < 0 && (((Im[B] Re[A] + Abs[Re[A]] [Sqrt](Im[B]^2 + 4 A Conjugate[A] Re[k]^2 - 4 Re[A + k Im[A] Im[B] + A Re[B] Re[k]]) < 2 Re[A] Re[ k Im[A] + A Im[k]] || (A Im[B] + Abs[A] [Sqrt](Im[B]^2 + 4 A (-1 - Re[B] Re[k] + A Re[k]^2)) > 2 A^2 Im[k] && A (2 A Im[ k] + [Sqrt](Im[B]^2 + 4 A (-1 - Re[B] Re[k] + A Re[k]^2)) Sign[A]) < A Im[B])) && (Sqrt[-4 A + Im[B]^2] == Re[B] || Sqrt[-4 A + Im[B]^2] + Re[B] == 0)) || ((Im[B] Re[A] + Abs[Re[A]] [Sqrt](Im[B]^2 + 4 A Conjugate[A] Re[k]^2 - 4 Re[A + k Im[A] Im[B] + A Re[B] Re[k]]) < 2 Re[A] Re[k Im[A] + A Im[k]] || Re[A] (2 Im[k] Re[A] + 2 Im[A] Re[ k] + [Sqrt](Im[B]^2 + 4 A Conjugate[A] Re[k]^2 - 4 Re[A + k Im[A] Im[B] + A Re[B] Re[k]]) Sign[ Re[A]]) < Im[B] Re[ A]) && ((((Im[A] Im[B])/Re[A] + Re[B] == ( Abs[A] Sqrt[Im[B]^2 - 4 Re[A]])/Abs[Re[A]] || Re[B] + (Im[A] Im[B] Sign[A] + A Sqrt[Im[B]^2 - 4 Re[A]] Sign[Re[A]])/(Re[ A] Sign[A]) == 0) && Abs[A] (A Conjugate[ B] + [Sqrt]((A + Conjugate[A]) ((8 A + B^2) Conjugate[A] + A Conjugate[B]^2)) + Conjugate[A] (B - 4 A Re[k])) != 0) || ((Sqrt[-4 A + Im[B]^2] < Re[B] || A^2 (Sqrt[-4 A + Im[B]^2] + Re[B]) < 0) && ((1/( A^2))(Sqrt[2] Abs[A] Sqrt[ 8 A + B^2 + Conjugate[B]^2] + 2 A Re[B]) < 4 Re[k] || Sqrt[2] A^2 Abs[A] Sqrt[ 8 A + B^2 + Conjugate[B]^2] + 4 A^4 Re[k] < 2 A^3 Re[B])) || (Sqrt[-4 A + Im[B]^2] > Re[B] && A^2 (Sqrt[-4 A + Im[B]^2] + Re[B]) > 0))) || (Im[B] Re[ A] + Abs[ Re[A]] [Sqrt](Im[B]^2 + 4 A Conjugate[A] Re[k]^2 - 4 Re[A + k Im[A] Im[B] + A Re[B] Re[k]]) != 2 Re[A] Re[ k Im[A] + A Im[k]] && (((1/( Abs[A]^2))([Sqrt]((A + Conjugate[A]) ((8 A + B^2) Conjugate[A] + A Conjugate[B]^2))) + 4 Re[k] == B/A + Conjugate[B]/ Conjugate[A] && ((Abs[A] Sqrt[Im[B]^2 - 4 Re[A]])/ Abs[Re[A]] < (Im[A] Im[B])/Re[A] + Re[B] || Re[A] (Im[A] Im[B] + Re[A] Re[ B] + (A Sqrt[Im[B]^2 - 4 Re[A]] Sign[Re[A]])/ Sign[A]) < 0)) || (Abs[ A] (A Conjugate[ B] + [Sqrt]((A + Conjugate[A]) ((8 A + B^2) Conjugate[A] + A Conjugate[B]^2)) + Conjugate[A] (B - 4 A Re[k])) == 0 && ((Abs[A] Sqrt[Im[B]^2 - 4 Re[A]])/ Abs[Re[A]] <= (Im[A] Im[B])/Re[A] + Re[B] || Re[B] + (Im[A] Im[B] Sign[A] + A Sqrt[Im[B]^2 - 4 Re[A]] Sign[Re[A]])/(Re[ A] Sign[A]) == 0 || Re[A] (Im[A] Im[B] + Re[A] Re[B] + (A Sqrt[Im[B]^2 - 4 Re[A]] Sign[Re[A]])/ Sign[A]) < 0)))) || (( A B + Sqrt[4 A + B^2] Abs[A])/A^2 > 2 Re[k] && A^2 Sqrt[4 A + B^2] Abs[A] + 2 A^4 Re[k] > A^3 B && (2 Sqrt[-A] < B || A^2 (2 Sqrt[-A] + B) < 0)))) || (A Im[B] + Abs[A] [Sqrt](Im[B]^2 + 4 A (-1 - Re[B] Re[k] + A Re[k]^2)) > 2 A^2 Im[k] && A (-Im[B] + 2 A Im[k] + [Sqrt](Im[B]^2 + 4 A (-1 - Re[B] Re[k] + A Re[k]^2)) Sign[A]) > 0 && ((A (Sqrt[2] Sqrt[8 A + B^2 + Conjugate[B]^2] - 2 Re[B] + 4 A Re[k]) < 0 && ((A >= 0 && 2 Sqrt[-A] != B) || A > 0)) || (A > 0 && ((4 A < Im[B]^2 && (((Sqrt[-4 A + Im[B]^2] == Re[B] || Sqrt[-4 A + Im[B]^2] + Re[B] == 0) && 4 A^2 Re[k] != A (B + Conjugate[B] + Sqrt[2] Sqrt[ 8 A + B^2 + Conjugate[B]^2])) || (Sqrt[-4 A + Im[B]^2] > Re[B] && A^2 (Sqrt[-4 A + Im[B]^2] + Re[B]) > 0))) || (A (B + Conjugate[B] + Sqrt[2] Sqrt[8 A + B^2 + Conjugate[B]^2] - 4 A Re[k]) < 0 && (2 (-A)^(5/2) < A^2 B || 2 I A^(5/2) + A^2 B < 0)))) || (A >= 0 && A (B + Conjugate[B] + Sqrt[2] Sqrt[8 A + B^2 + Conjugate[B]^2] - 4 A Re[k]) < 0))))) || (B == Re[B] && ((Re[A] < 0 && (((Abs[ Re[A]] [Sqrt](A Conjugate[A] Re[k]^2 - Re[A] (1 + Re[B] Re[k])) < Re[A] Re[k Im[A] + A Im[k]] || (Im[k] != Sqrt[ A (-1 - Re[B] Re[k] + A Re[k]^2)]/Abs[A] &&

           Re[A] (Im[k] Re[A] + 
               Im[A] Re[
                k] + \[Sqrt](A Conjugate[A] Re[k]^2 - 
                Re[A] (1 + Re[B] Re[k])) Sign[Re[A]]) < 
            0)) && (2 Sqrt[-A] == Re[B] || 
         2 Sqrt[-A] + Re[B] == 
          0)) || ((Abs[
            Re[A]] \[Sqrt](A Conjugate[A] Re[k]^2 - 
              Re[A] (1 + Re[B] Re[k])) < 
          Re[A] Re[k Im[A] + A Im[k]] || 
         Re[A] (Im[k] Re[A] + 
             Im[A] Re[
               k] + \[Sqrt](A Conjugate[A] Re[k]^2 - 
                Re[A] (1 + Re[B] Re[k])) Sign[Re[A]]) < 
          0) && (((2 Sqrt[-((A Conjugate[A])/Re[A])] == Re[B] || 
             2 Sqrt[-((A Conjugate[A])/Re[A])] + Re[B] == 0) && (
            1/(Abs[A]^2))(Re[A] Re[
                B] + \[Sqrt](Re[
                A] (4 A Conjugate[A] + Re[A] Re[B]^2))) != 
            2 Re[k]) || (2 Sqrt[-A] > Re[B] && 
           2 Sqrt[-A] + Re[B] > 0))) || (\[Sqrt]((1/(
          Re[A]^2))(A Conjugate[A] Re[k]^2 - 
            Re[A] (1 + Re[B] Re[k]))) != 
        Im[k] + (Im[A] Re[k])/
         Re[A] && (((1/(
            Abs[A]^2))(Re[A] Re[
                B] + \[Sqrt](Re[
                A] (4 A Conjugate[A] + Re[A] Re[B]^2))) == 
            2 Re[k] && (2 Sqrt[-((A Conjugate[A])/Re[A])] <= 
              Re[B] || 
             2 Sqrt[-((A Conjugate[A])/Re[A])] + Re[B] <= 
              0)) || ((1/
            A)(-Re[A] Re[
                B] + \[Sqrt](Re[
                A] (4 A Conjugate[A] + Re[A] Re[B]^2)) + 
               2 A Conjugate[A] Re[k]) Sign[A] == 
            0 && (2 Sqrt[-((A Conjugate[A])/Re[A])] < Re[B] || 
             2 Sqrt[-((A Conjugate[A])/Re[A])] + Re[B] < 
              0)))) || ((((Abs[
                Re[A]] \[Sqrt](A Conjugate[A] Re[k]^2 - 
                Re[A] (1 + Re[B] Re[k])) < 
              Re[A] Re[k Im[A] + A Im[k]] || 
             Re[A] (Im[k] Re[A] + 
                Im[A] Re[
                k] + \[Sqrt](A Conjugate[A] Re[k]^2 - 
                Re[A] (1 + Re[B] Re[k])) Sign[Re[A]]) < 0) && ((
              A Re[B] + Abs[A] Sqrt[4 A + Re[B]^2])/A^2 < 
              2 Re[k] || 
             A^2 Abs[A] Sqrt[4 A + Re[B]^2] + 2 A^4 Re[k] < 
              A^3 Re[B])) || ((
            A Re[B] + Abs[A] Sqrt[4 A + Re[B]^2])/A^2 > 2 Re[k] &&
            A^2 Abs[A] Sqrt[4 A + Re[B]^2] + 2 A^4 Re[k] > 
            A^3 Re[B])) && (2 Sqrt[-A] < Re[B] || 
         2 Sqrt[-A] + Re[B] < 0)))) || (A > 0 && 
   A Sqrt[A (-1 - Re[B] Re[k] + A Re[k]^2)] > A^2 Im[k] && 
   A (A Im[k] + Sqrt[A (-1 - Re[B] Re[k] + A Re[k]^2)]) > 
    0 && (A (Re[B] + Sqrt[4 A + Re[B]^2] - 2 A Re[k]) < 0 || 
     A (-Re[B] + Sqrt[4 A + Re[B]^2] + 2 A Re[k]) < 0))))`
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