Sum

Sum[f,{i,imax}]
evaluates the sum .

Sum[f,{i,imin,imax}]
starts with i=i_(min).

Sum[f,{i,imin,imax,di}]
uses steps di.

Sum[f,{i,{i1,i2,}}]
uses successive values i_(1), i_(2), .

Sum[f,{i,imin,imax},{j,jmin,jmax},]
evaluates the multiple sum .

Sum[f,i]
gives the indefinite sum .

Details and OptionsDetails and Options

  • Sum[f,{i,imax}] can be entered as sum_(i)^(i_(max))f.
  • sum can be entered as EscsumEsc or \[Sum].
  • Sum[f,{i,imin,imax}] can be entered as sum_(i=i_(min))^(i_(max))f.
  • The limits should be underscripts and overscripts of sum in normal input, and subscripts and superscripts when embedded in other text.
  • Sum uses the standard Wolfram Language iteration specification.
  • The iteration variable i is treated as local, effectively using Block.
  • If the range of a sum is finite, is typically assigned a sequence of values, with being evaluated for each one.
  • In multiple sums, the range of the outermost variable is given first. »
  • The limits of summation need not be numbers. They can be Infinity or symbolic expressions. » »
  • If a sum cannot be carried out explicitly by adding up a finite number of terms, Sum will attempt to find a symbolic result. In this case, f is first evaluated symbolically.
  • The indefinite sum is defined so that its difference with respect to i gives f. »
  • Definite and indefinite summation can be mixed in any order. »
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate conditions on parameters
    MethodAutomaticmethod to use
    RegularizationNonewhat regularization scheme to use
    VerifyConvergenceTruewhether to verify convergence
  • Possible values for Regularization include: None, "Abel", "Borel", "Cesaro", "Dirichlet", and "Euler". {reg1,reg2,} specifies different schemes for different variables in a multiple sum.
  • Method->"method" performs the summation using the specified method.
  • Method->{"strategy",Method->{"meth1","meth2",}} uses the methods "methi", controlled by the specified strategy method.
  • Possible strategy methods include:
  • "SequentialFirstToSucceed"sequentially try each method until one succeeds
    "SequentialBestQuality"sequentially try each method and return the best result
    "ParallelFirstToSucceed"try each method in parallel until one succeeds
    "ParallelBestQuality"try each method in parallel and return the best result
    "IteratedSummation"use iterated univariate summation
  • Specific methods include:
  • Automaticautomatically selected method
    "HypergeometricTermFinite"special finite hypergeometric term summation
    "HypergeometricTermGosper"indefinite hypergeometric term summation
    "HypergeometricTermPFQ"general definite hypergeometric term summation
    "HypergeometricTermZeilberger"definite hypergeometric term summation
    "LevelCounting"summation based on counting solutions in level sets
    "Logarithmic"logarithmic series summation
    "PeriodicFunction"periodic function summation
    "PolyGammaHypergeometricSeries"polygamma series representation summation
    "PolyGammaIntegralRepresentation"polygamma integral representation summation
    "PolyGammaSumByParts"polygamma summation by parts
    "Polynomial"polynomial summation
    "PolynomialExponential"polynomial exponential summation
    "PolynomialTrigonometric"polynomial trigonometric summation
    "Procedural"compute the sum procedurally
    "QHypergeometricTermGosper"indefinite q-hypergeometric term summation
    "QHypergeometricTermZeilberger"definite q-hypergeometric term summation
    "QRationalFunction"q-rational function summation
    "RationalExponential"rational times exponential summation
    "RationalFunction"rational function summation
    "RationalTrigonometric"rational trigonometric summation
    "TableLookup"summation based on table lookup
  • Sum can do essentially all sums that are given in standard books of tables.
  • Sum is output in StandardForm using .

ExamplesExamplesopen allclose all

Basic Examples  (6)Basic Examples  (6)

Numeric sum:

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Symbolic sum:

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Use EscsumEsc to enter and Ctrl+_ for the lower limit and then Ctrl+% for the upper limit:

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Infinite sum:

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Indefinite sum:

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Multiple sum with summation over j performed first:

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Introduced in 1988
(1.0)
| Updated in 2008
(7.0)
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