AsymptoticSum

AsymptoticSum[f,i,{i,,n}]

computes an asymptotic approximation of the indefinite sum for i centered at of order n.

AsymptoticSum[f,{i,a,b},{α,α0,n}]]

computes an asymptotic approximation of the definite sum for α centered at α0 of order n.

Details and Options

  • Asymptotic approximations to sums are also known as asymptotic expansions and perturbation expansions. They are also known by specific methods to compute some of them, such as the EulerMaclaurin method, summation by parts, etc.
  • Asymptotic approximations are typically used to compute sums for which no exact result can be found or to get simpler answers for computation, comparison and interpretation.
  • If the exact result is g[x] and the asymptotic approximation of order n at x0 is gn[x], then AsymptoticLess[g[x]-gn[x],gn[x]-gn-1[x],xx0] or g[x]-gn[x]o[gn[x]-gn-1[x]] as xx0.
  • The asymptotic approximation gn[x] is often given as a sum gn[x]αkϕk[x], where {ϕ1[x],,ϕn[x]} is an asymptotic scale ϕ1[x]ϕ2[x]>ϕn[x] as xx0. Then AsymptoticLess[g[x]-gn[x],ϕn[x],xx0] or g[x]-gn[x]o[ϕn[x]] as xx0.
  • Taylor scale when xx0
    Laurent scale when xx0
    Laurent scale when x±
    Puiseaux scale when xx0
  • The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
  • The center α0 can be any finite or infinite real or complex number.
  • The order n must be a positive integer and specifies order of approximation for the asymptotic expansion. It is not related to polynomial degree.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
    GeneratedParametersNonehow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticSum typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Compute an asymptotic expansion for a sum:

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Compare with the exact value for large n:

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Compute an asymptotic expansion for a sum with respect to a parameter:

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Compute the required expansion:

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Compare with the exact value for small a:

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Scope  (13)

Options  (1)

Applications  (9)

Properties & Relations  (3)

Introduced in 2019
(12.0)