SumConvergence

SumConvergence[f,n]

gives conditions for the sum to be convergent.

SumConvergence[f,{n1,n2,…}]

gives conditions for the multiple sum to be convergent.

SumConvergence[f,{n,a,∞}]

gives conditions for the sum to be convergent on the interval .

SumConvergence[f,{n,a,∞},…,{m,b,∞}]

gives conditions for the multiple sum to be convergent.

Details and Options

  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    Direction1direction of summation
    Method Automaticmethod to use for convergence testing
  • Possible values for Method include:
  • "IntegralTest"the integral test
    "RaabeTest"Raabe's test
    "RatioTest"D'Alembert ratio test
    "RootTest"Cauchy root test
  • With the default setting Method->Automatic, a number of additional tests specific to different classes of sequences are used.
  • For multiple sums, convergence tests are performed for each independent variable.

Examples

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Basic Examples  (4)

Test for convergence of the sum :

Test the convergence of :

Find the condition for convergence of :

Test for convergence of the sum :

Find the conditions for convergence of :

Scope  (16)

Numerical Sums  (8)

Exponential or geometric sums:

Plot the partial sums:

Polynomial exponential sums:

Rational sums:

Convergence picture:

Special functions:

Piecewise functions:

Slowly converging sums in the Abel–Dini scale:

Alternating sums:

Complex-valued sums:

Parametric Sums  (6)

Exponential or geometric series:

Parameter region for convergence:

Power series:

The convergence region for :

Combined series:

Piecewise sums:

Assuming z=u+ v to be complex:

A multivariate sum:

Convergence on Intervals  (2)

Test the convergence of on different intervals:

Test the convergence of a parametric sum on different intervals:

Options  (10)

Method  (10)

Test the convergence of using the ratio test:

Test the convergence of using the ratio test:

In this case the ratio test is inconclusive:

Test the convergence of using the root test:

Test the convergence of using the root test:

In this case the root test is inconclusive:

The Raabe test works well for rational functions:

In this case the Raabe test is inconclusive:

Test the convergence of using the integral test:

Test the convergence of using the integral test:

In this case the integral test is inconclusive:

Applications  (3)

Find the radius of convergence of a power series:

Find the interval of convergence for a real power series:

As a real power series, this converges on the interval [-3,3):

Prove convergence of Ramanujan's formula for :

Sum it:

Properties & Relations  (4)

Convergence properties are not affected by multiplication of constants:

Convergence is not affected by translating arguments:

SumConvergence is automatically called by Sum:

Many conditions generated by Sum are in effect convergence conditions:

With the setting VerifyConvergence->False, typically a regularized value is returned:

SumConvergence is used in sum transforms such as ZTransform:

GeneratingFunction:

ExponentialGeneratingFunction:

FourierSequenceTransform:

Neat Examples  (1)

Conditionally convergent periodic sums:

Wolfram Research (2008), SumConvergence, Wolfram Language function, https://reference.wolfram.com/language/ref/SumConvergence.html (updated 2025).

Text

Wolfram Research (2008), SumConvergence, Wolfram Language function, https://reference.wolfram.com/language/ref/SumConvergence.html (updated 2025).

CMS

Wolfram Language. 2008. "SumConvergence." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/SumConvergence.html.

APA

Wolfram Language. (2008). SumConvergence. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SumConvergence.html

BibTeX

@misc{reference.wolfram_2025_sumconvergence, author="Wolfram Research", title="{SumConvergence}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/SumConvergence.html}", note=[Accessed: 23-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_sumconvergence, organization={Wolfram Research}, title={SumConvergence}, year={2025}, url={https://reference.wolfram.com/language/ref/SumConvergence.html}, note=[Accessed: 23-March-2025 ]}