gives conditions for the sum to be convergent.


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
    MethodAutomaticmethod 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.


open allclose all

Basic Examples  (2)

Test for convergence of the sum :

Test the convergence of :

Find the condition for convergence of :

Scope  (14)

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 AbelDini 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:

Options  (9)

Method  (9)

In this case the ratio test is inconclusive:

In this case the root test is inconclusive:

The Raabe test works well for rational functions:

In this case the Raabe test is inconclusive:

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:




Neat Examples  (1)

Conditionally convergent periodic sums:

Wolfram Research (2008), SumConvergence, Wolfram Language function, (updated 2010).


Wolfram Research (2008), SumConvergence, Wolfram Language function, (updated 2010).


@misc{reference.wolfram_2020_sumconvergence, author="Wolfram Research", title="{SumConvergence}", year="2010", howpublished="\url{}", note=[Accessed: 17-January-2021 ]}


@online{reference.wolfram_2020_sumconvergence, organization={Wolfram Research}, title={SumConvergence}, year={2010}, url={}, note=[Accessed: 17-January-2021 ]}


Wolfram Language. 2008. "SumConvergence." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010.


Wolfram Language. (2008). SumConvergence. Wolfram Language & System Documentation Center. Retrieved from