NumericalCalculus`

EulerSum

EulerSum[f,{i,i_min,i_max}]

gives a numerical approximation to the sum f using Euler's transformation.

EulerSum[f,{i,i_min,i_max,di}]

uses a step di in the sum.

Details and Options

To use EulerSum, you first need to load the Numerical Calculus Package using Needs["NumericalCalculus`"].
EulerSum is particularly good for working with alternating series and series whose terms have the form , where is a polynomial in .
EulerSum can be used for sums with both finite and infinite limits.
The following options can be given:

EulerRatio	Automatic	parameter used in generalized Euler transformations
ExtraTerms	7	number of terms to use during extrapolation
Terms	5	number of terms to use before extrapolation
WorkingPrecision	MachinePrecision	the 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.

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