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EulerSum

EulerSum[f, {i, imin, imax}]
gives a numerical approximation to the sum f using Euler's transformation.
EulerSum[f, {i, imin, imax, di}]
uses a step di in the sum.
  • EulerSum is particularly good for working with alternating series and series whose terms have the form p(n) r^n, where p(n) is a polynomial in n.
  • EulerSum can be used for sums with both finite and infinite limits.
  • The following options can be given:
EulerRatioAutomaticparameter used in generalized Euler transformations
ExtraTerms7number of terms to use during extrapolation
Terms5number of terms to use before extrapolation
WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • Valid settings for EulerRatio are Automatic, a single ratio, or a list of ratios or {ratio, degree+1} pairs. In the case of a list, the various ratios are used successively in iterated Euler transformations.
  • With the default setting Automatic, the value used for EulerRatio is based on the ratio of successive terms in the series.
  • Valid settings for ExtraTerms are integers greater than 1. The default setting is 7.
  • The setting for ExtraTerms gives the number of terms to be used in the extrapolation process after the number of terms specified by the option Terms have been explicitly summed.
  • Valid settings for Terms are positive integers.