Function Repository Resource:

# CRVZSum

Evaluate an alternating sum using the Cohen-Rodriguez Villegas-Zagier method

Contributed by: Jan Mangaldan
 ResourceFunction["CRVZSum"][f,{i,imin,∞}] numerically evaluates the sum using the CRVZ algorithm.

## Details and Options

The Cohen-Rodriguez Villegas-Zagier (CRVZ) algorithm is an acceleration technique intended for infinite alternating sums.
The following options can be given:
 "ExtraTerms" 15 number of terms to use in the CRVZ method "Terms” 15 number of terms to sum directly WorkingPrecision MachinePrecision the precision used in internal computations
Given sufficiently pathological summands, ResourceFunction["CRVZSum"] can give wrong answers. Generally one can test the answer by looking at its sensitivity to changes in the setting of options for ResourceFunction["CRVZSum"].
ResourceFunction["CRVZSum"] has the attribute HoldAll and effectively uses Block to localize variables.

## Examples

### Basic Examples (2)

Evaluate the alternating harmonic series:

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Compare with the closed form:

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### Options (5)

#### ExtraTerms (2)

Use 25 terms for the CRVZ extrapolation:

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Compare with the exact result:

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Set "Terms" to 0 so that all terms are used in extrapolation:

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Compare with the exact result:

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#### Terms (2)

Directly sum the first 25 terms before applying CRVZ extrapolation:

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Compare with the exact result:

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#### WorkingPrecision (1)

Use a higher setting of WorkingPrecision:

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### Applications (2)

Use CRVZSum to evaluate the Dirichlet eta function:

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Compare with the built-in DirichletEta:

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Plot the relative error:

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Use CRVZSum with NIntegrate to numerically evaluate an oscillatory integral:

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Compare with the exact result:

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### Properties and Relations (2)

Directly summing the first few terms of an alternating series usually does not give sufficient accuracy:

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Using the CRVZ method on an alternating series often gives better results:

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### Possible Issues (2)

CRVZSum usually gives poor results for non-alternating series:

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CRVZSum may give results for formally divergent series:

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Compare with the Borel sum:

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## Version History

• 1.0.0 – 29 March 2021