gives a numerical approximation to the sum .


uses a step di in the sum.

Details and Options

  • NSum can be used for sums with both finite and infinite limits.
  • NSum[f,{i,},{j,},] can be used to evaluate multidimensional sums.
  • The following options can be given:
  • AccuracyGoalInfinitynumber of digits of final accuracy sought
    EvaluationMonitorNoneexpression to evaluate whenever f is evaluated
    MethodAutomaticmethod to use
    NSumTerms15number of terms to use before extrapolation
    PrecisionGoalAutomaticnumber of digits of final precision sought
    VerifyConvergenceTruewhether to explicitly test for convergence
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • Possible settings for the Method option include:
  • "AlternatingSigns"method for summands with alternating signs
    "EulerMaclaurin"EulerMaclaurin summation method
    "WynnEpsilon"Wynn epsilon extrapolation method
  • With the EulerMaclaurin method, the options AccuracyGoal and PrecisionGoal can be used to specify the accuracy and precision to try and get in the final answer. NSum stops when the error estimates it gets imply that either the accuracy or precision sought has been reached.
  • You should realize that with sufficiently pathological summands, the algorithms used by NSum can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NSum.
  • VerifyConvergence is only used for sums with infinite limits.
  • N[Sum[]] calls NSum for sums that cannot be done symbolically.
  • NSum first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • NSum has attribute HoldAll, and effectively uses Block to localize variables.


open allclose all

Basic Examples  (1)

Approximate an infinite sum numerically:

Click for copyable input

The error versus the exact value of :

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Scope  (5)

Options  (9)

Applications  (2)

Properties & Relations  (3)

Possible Issues  (3)

See Also

Sum  NProduct  Total  NIntegrate


Introduced in 1988
| Updated in 2007