AsymptoticSum

AsymptoticSum[f,x,xx0]

computes an asymptotic approximation of the indefinite sum for x centered at x0.

AsymptoticSum[f,{x,a,b},αα0]

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

AsymptoticSum[f,,{ξ,ξ0,n}]

computes the asymptotic approximation to 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.
  • AsymptoticSum[f,,xx0] computes the leading term in an asymptotic expansion for the sum of f. Use SeriesTermGoal to specify more terms.
  • 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:
  • AccuracyGoalAutomaticdigits of absolute accuracy sought
    Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
    GeneratedParametersNonehow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoalAutomaticdigits of precision sought
    RegularizationNonewhat regularization scheme to use
    SeriesTermGoalAutomaticnumber of terms in the approximation
    WorkingPrecisionAutomaticthe precision used in internal computations
  • 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  (3)

Compute a leading asymptotic approximation for a sum:

Compute an asymptotic expansion for a sum:

Compare with the exact value for large n:

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

Compute the required expansion:

Compare with the exact value for small a:

Scope  (13)

Indefinite Sums  (5)

Compute an asymptotic expansion for a rational sum:

Compare with the result given by Sum:

Asymptotic expansion for a polynomial-exponential sum:

Compare with the result given by Sum:

Asymptotic expansion for a hypergeometric sum:

Estimate the value at a point:

Compare with the value given by NSum:

Asymptotic expansion for a rational-exponential sum:

Verify the result:

Asymptotic expansion for a PolyGamma sum:

Definite Sums  (4)

Compute an asymptotic expansion for a rational sum:

Compare with the result given by NSum:

Asymptotic expansion for a rational-exponential sum:

Compare with the result given by NSum:

Compute an asymptotic expansion for a hypergeometric sum:

Compare with the exact result:

Asymptotic expansion for a HarmonicNumber sum:

Compare with the result given by NSum:

Parametric Sums  (4)

Compute an asymptotic expansion with respect to a parameter:

Verify the result:

Compute an asymptotic expansion for an infinite exponential sum:

Compare with the exact result:

Asymptotic expansion for a sum related to Zeta:

Compare with a numerical approximation:

Asymptotic expansion for an alternating Gaussian exponential sum:

Compare with a numerical approximation:

Options  (3)

GeneratedParameters  (1)

Generate an arbitrary constant for an indefinite sum:

The default value for the arbitrary constant is 0:

Regularization  (2)

Compute an asymptotic expansion for a divergent sum using Abel regularization:

Compute an asymptotic expansion for a divergent sum using Borel regularization:

Applications  (9)

Compute an approximation for an finite sum:

Compute the numerical approximation for increasing values of n:

Compare with the exact results given by Sum:

Compute a Riemann sum approximation for a definite integral:

Compute the approximation for large values of n:

Compare with the result given by Integrate:

Obtain the exact result using DiscreteLimit:

Plot the approximations and the exact value:

Compute a numerical approximation for an improper integral:

Compute the approximation for large values of n:

Compare with the result given by NIntegrate:

Show that :

Compare with an expansion of the exact result:

Compute an asymptotic expansion for :

Compare the approximate and exact values for :

Compute an asymptotic approximation for a Gaussian sum:

Compare the approximate and exact values for :

Compute the first-order asymptotic approximation for a rational sum:

Rational-exponential sum:

Hypergeometric sum:

Show that is asymptotically equivalent to 1/n, as n approaches Infinity:

Verify the required equivalence using AsymptoticEquivalent:

Verify the result using the result from Sum:

Compute an asymptotic approximation for a binomial sum:

Compare the approximate and exact values for :

Properties & Relations  (3)

AsymptoticSum computes the sum up to a given order:

Use Sum to compute the sum in closed form:

Use NSum to compute a numerical approximation:

Introduced in 2019
 (12.0)
 |
Updated in 2020
 (12.1)