Sum

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

Sum[f,{i,imin,imax}]
starts with .

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

Sum[f,{i,{i1,i2,}}]
uses successive values , , .

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 .
  • can be entered as EscsumEsc or \[Sum].
  • Sum[f,{i,imin,imax}] can be entered as .
  • The limits should be underscripts and overscripts of 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, , , , , and . 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 , controlled by the specified strategy method.
  • Possible strategy methods include:
  • sequentially try each method until one succeeds
    sequentially try each method and return the best result
    try each method in parallel until one succeeds
    try each method in parallel and return the best result
    use iterated univariate summation
  • Specific methods include:
  • Automaticautomatically selected method
    special finite hypergeometric term summation
    indefinite hypergeometric term summation
    general definite hypergeometric term summation
    definite hypergeometric term summation
    summation based on counting solutions in level sets
    logarithmic series summation
    periodic function summation
    polygamma series representation summation
    polygamma integral representation summation
    polygamma summation by parts
    polynomial summation
    polynomial exponential summation
    polynomial trigonometric summation
    compute the sum procedurally
    indefinite q-hypergeometric term summation
    definite q-hypergeometric term summation
    q-rational function summation
    rational times exponential summation
    rational function summation
    rational trigonometric summation
    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 .
Introduced in 1988
(1.0)
| Updated in 2008
(7.0)