Revisions to Getting wrong limit with Bessel

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edited Jan 19, 2021 at 13:37

J. M.'s missing motivation ♦

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Bug introduced in 7.0 or earlier and persisting through 11.1 or later. Fixedfixed in 11.3

The bug was fixed in version 11.3

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edited Mar 11, 2018 at 23:26

Vaclav Kotesovec

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Bug introduced in 7.0 or earlier and persisting through 11.1 or later. Fixed in 11.3

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edited Apr 7, 2017 at 7:40

J. M.'s missing motivation ♦

125.5k
11
407
581

Bug introduced in 7.0 or earlier and persisting through 11.1 or later

I computed a following limit (related to the asymptotic expansion of the sequence A000009 - number of partitions of nn into distinct parts)

Expand[Limit[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/
             (E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) -
             (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] +
              (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), n -> Infinity]]

MathematicaMathematica wrong output (in all versions from 7 to 11) is

(* -((315 Sqrt[3])/(1024 π^3)) + (35 Sqrt[3])/(2048 π) -
   (35 π)/(36864 Sqrt[3]) + π^3/(1990656 Sqrt[3]) *)

My question is: Whywhy is the term

missing in Mathematicathe Mathematica result? This is a bug!

Interesting is that numerically Mathematica evaluateMathematica evaluates a following expression correctly:

Table[N[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/
         (E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) -
         (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] +
          (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), 10],
      {n, 1000000, 10000000, 1000000}]

(* {-0.009484688338, -0.009481807906, -0.009480532562, -0.009479772520, -0.009479253935, -0.009478871178, -0.009478573728, -0.009478333979, -0.009478135403, -0.009477967422} *)

I computed a following limit (related to the asymptotic expansion of the sequence A000009 - number of partitions of n into distinct parts)

Expand[Limit[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/(E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) - (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] + (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), n -> Infinity]]

Mathematica wrong output (in all versions from 7 to 11) is

(* -((315 Sqrt[3])/(1024 π^3)) + (35 Sqrt[3])/(2048 π) - (35 π)/(36864 Sqrt[3]) + π^3/(1990656 Sqrt[3]) *)

My question is: Why is the term

missing in Mathematica result? This is a bug!

Interesting is that numerically Mathematica evaluate a following expression correctly:

Table[N[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/(E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) - (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] + (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), 10], {n, 1000000, 10000000, 1000000}]

(* {-0.009484688338, -0.009481807906, -0.009480532562, -0.009479772520, -0.009479253935, -0.009478871178, -0.009478573728, -0.009478333979, -0.009478135403, -0.009477967422} *)

Bug introduced in 7.0 or earlier and persisting through 11.1 or later

I computed a following limit (related to the asymptotic expansion of the sequence A000009 - number of partitions of n into distinct parts)

Expand[Limit[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/
             (E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) -
             (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] +
              (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), n -> Infinity]]

Mathematica wrong output (in all versions from 7 to 11) is

(* -((315 Sqrt[3])/(1024 π^3)) + (35 Sqrt[3])/(2048 π) -
   (35 π)/(36864 Sqrt[3]) + π^3/(1990656 Sqrt[3]) *)

My question is: why is the term

missing in the Mathematica result? This is a bug!

Interesting is that numerically Mathematica evaluates a following expression correctly:

Table[N[(((π BesselI[1, (Sqrt[1/24 + n] π)/Sqrt[3]])/Sqrt[1 + 24 n])/
         (E^((Sqrt[n] π)/Sqrt[3])/(4 3^(1/4) n^(3/4))) -
         (1 + (-((3 Sqrt[3])/(8 π)) + π/(48 Sqrt[3]))/Sqrt[n] +
          (-(5/128) - 45/(128 π^2) + π^2/13824)/n))*n^(3/2), 10],
      {n, 1000000, 10000000, 1000000}]

(* {-0.009484688338, -0.009481807906, -0.009480532562, -0.009479772520, -0.009479253935, -0.009478871178, -0.009478573728, -0.009478333979, -0.009478135403, -0.009477967422} *)