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SumConvergence
✖
SumConvergence

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✖

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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Test for convergence of the sum :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-c1vet4
Out[1]=1

Test the convergence of :

In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-etel5q
Out[2]=2

Find the condition for convergence of :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-cu8qnm
Out[1]=1

Test for convergence of the sum :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-n63cu8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-l4rbl
Out[2]=2

Find the conditions for convergence of :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-pwc8dd
Out[1]=1

Scope  (16)Survey of the scope of standard use cases

Numerical Sums  (8)

Exponential or geometric sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-dyjp3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-e1xeg3
Out[2]=2

Plot the partial sums:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-bu29o
Out[3]=3

Polynomial exponential sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-ezajut
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-lxetf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-cw6l87
Out[3]=3

Rational sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-kwwl5d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-clb72l
Out[2]=2

Convergence picture:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-eoy08d
Out[3]=3

Special functions:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-ma1e3t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-hwkxs
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-ec33kz
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dczi5kbwbe0-4ktai
Out[4]=4

Piecewise functions:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-fc3uot
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-cqabea
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-cul2fe
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dczi5kbwbe0-usgli
Out[4]=4

Slowly converging sums in the Abel–Dini scale:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-fdp34p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-fqfqaj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-dtwl00
Out[3]=3

Alternating sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-4of3u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-7fqh7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-ejaw00
Out[3]=3

Complex-valued sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-eqcww1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-174ju
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-f8ag6s
Out[3]=3

Parametric Sums  (6)

Exponential or geometric series:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-d6jwn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-99t9n
Out[2]=2

Parameter region for convergence:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-bg7t6f
Out[3]=3

Power series:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-jnb4im
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-dtdyst
Out[2]=2

The convergence region for :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-bdx3fh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-bazhqg
Out[2]=2

Combined series:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-kbsor2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-h7mf7j
Out[2]=2

Piecewise sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-dzhaqa
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-oz525y
Out[2]=2

Assuming z=u+ v to be complex:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-bimlvy
Out[3]=3

A multivariate sum:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-enrjmh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-bmm5vg
Out[2]=2

Convergence on Intervals  (2)

Test the convergence of on different intervals:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-btd3if
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-il5bzn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-hti5tk
Out[3]=3

Test the convergence of a parametric sum on different intervals:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-hf2p7k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-cd5xw9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-ghjs6a
Out[3]=3

Options  (10)Common values & functionality for each option

Method  (10)

Test the convergence of using the ratio test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-eypfh5
Out[1]=1

Test the convergence of using the ratio test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-tqf69
Out[1]=1

In this case the ratio test is inconclusive:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-2x2lm
Out[1]=1

Test the convergence of using the root test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-h16r6a
Out[1]=1

Test the convergence of using the root test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-bct9ub
Out[1]=1

In this case the root test is inconclusive:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-bw3dpp
Out[1]=1

The Raabe test works well for rational functions:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-d9314m
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-cafs6g
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-erz0cz
Out[3]=3

In this case the Raabe test is inconclusive:

In[4]:=4
✖
https://wolfram.com/xid/0dczi5kbwbe0-j0xmsf
Out[4]=4

Test the convergence of using the integral test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-cwsmkp
Out[1]=1

Test the convergence of using the integral test:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-0nm8g
Out[1]=1

In this case the integral test is inconclusive:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-c7n6eb
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Find the radius of convergence of a power series:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-dd789c
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-gcl8d9
Out[1]=1

Find the interval of convergence for a real power series:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-nbikk
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-b2j8y2
Out[2]=2

Prove convergence of Ramanujan's formula for :

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-khp1q
Out[1]=1

Sum it:

In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-htqump
Out[2]=2

Properties & Relations  (4)Properties of the function, and connections to other functions

Convergence properties are not affected by multiplication of constants:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-jkc0iz
Out[1]=1

Convergence is not affected by translating arguments:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-kbcl13
Out[1]=1

SumConvergence is automatically called by Sum:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-c05zy7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-dh6nxx
Out[2]=2

Many conditions generated by Sum are in effect convergence conditions:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-eparh
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dczi5kbwbe0-9c5nv
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0dczi5kbwbe0-bo5xgi
Out[5]=5

SumConvergence is used in sum transforms such as ZTransform:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-fnpvhk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-bxe6pt
Out[2]=2

GeneratingFunction:

In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-q4rny3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dczi5kbwbe0-iizen3
Out[4]=4

ExponentialGeneratingFunction:

In[5]:=5
✖
https://wolfram.com/xid/0dczi5kbwbe0-d9gqo7
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dczi5kbwbe0-bu7gfj
Out[6]=6

FourierSequenceTransform:

In[7]:=7
✖
https://wolfram.com/xid/0dczi5kbwbe0-bbvqpz
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0dczi5kbwbe0-gesdws
Out[8]=8

Neat Examples  (1)Surprising or curious use cases

Conditionally convergent periodic sums:

In[1]:=1
✖
https://wolfram.com/xid/0dczi5kbwbe0-l16xn
In[2]:=2
✖
https://wolfram.com/xid/0dczi5kbwbe0-hwhdut
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dczi5kbwbe0-hxmfaw
Out[3]=3
Wolfram Research (2008), SumConvergence, Wolfram Language function, https://reference.wolfram.com/language/ref/SumConvergence.html (updated 2025).
✖
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).

✖
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.

✖
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

✖
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: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_sumconvergence, author="Wolfram Research", title="{SumConvergence}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/SumConvergence.html}", note=[Accessed: 29-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: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_sumconvergence, organization={Wolfram Research}, title={SumConvergence}, year={2025}, url={https://reference.wolfram.com/language/ref/SumConvergence.html}, note=[Accessed: 29-March-2025 ]}