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.
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 $Assumptions assumptions to make about parameters Direction 1 direction of summation Method Automatic method 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 all close allBasic Examples (4)
Test for convergence of the sum 
:
SumConvergence[1 / n, n]
SumConvergence[3 ^ n n ^ 2 / n!, n]Find the condition for convergence of 
:
SumConvergence[1 / n ^ α, n]Test for convergence of the sum 
:
SumConvergence[1 / n ^ 2, {n, 1, ∞}]SumConvergence[1 / n ^ 2, {n, 0, ∞}]Find the conditions for convergence of 
:
SumConvergence[1 / (n - a) ^ 2, {n, 5, Infinity}]Scope (16)
Numerical Sums (8)
Exponential or geometric sums:
SumConvergence[(-1 / 2) ^ n, n]SumConvergence[2 ^ n, n]DiscretePlot[Sum[(-1 / 2) ^ n, {n, 0, k}], {k, 25}, PlotRange -> All, AxesOrigin -> {0, 0}]SumConvergence[(1 / 2) ^ n n ^ 3, n]SumConvergence[(1 / 2) ^ n n ^ 10, n]DiscretePlot[Sum[(1 / 2) ^ n n ^ 10, {n, 0, k}], {k, 50}]SumConvergence[1 / n, n]SumConvergence[(n + 3) / ((n ^ 2 + 2)(n + 1)), n]{DiscretePlot[Sum[1 / n, {n, 1, k}], {k, 25}], DiscretePlot[Sum[(n + 3) / ((n ^ 2 + 2)(n + 1)), {n, 1, k}], {k, 1, 25}]}SumConvergence[HarmonicNumber[n] / ((2n + 1)(3n + 1)), n]SumConvergence[PolyGamma[n] / n!, n]SumConvergence[Zeta[n] (-1) ^ n / Log[n], n]DiscretePlot[Sum[Zeta[n] (-1) ^ n / Log[n], {n, 2, k}], {k, 25}]SumConvergence[Max[1 / n, 1 / n ^ 2] + UnitStep[7 - n] / n ^ 4, n]SumConvergence[(1/n^2 + Floor[(n/1 + n) + Floor[(12/1 + n^2)]]), n]SumConvergence[Boole[n ^ 2 < 1000]n, n]DiscretePlot[Sum[Boole[n ^ 2 < 1000] n, {n, k}], {k, 50}]Slowly converging sums in the Abel–Dini scale:
SumConvergence[1 / (n Log[n] Log[Log[n]]), n]SumConvergence[1 / (n Log[n]Log[Log[n]] ^ 2), n]DiscretePlot[Sum[1 / (n Log[n]Log[Log[n]] ^ 2), {n, 10, k}], {k, 10, 100}]SumConvergence[(-1) ^ n, n]SumConvergence[(-1) ^ n / (2n + 1), n]DiscretePlot[Sum[(-1) ^ n / (2n + 1), {n, 0, k}], {k, 25}]SumConvergence[Exp[n 2Pi I / 8] / n, n]SumConvergence[1 / (n ^ 2 + I n + 2I), n]{DiscretePlot[s = Sum[Exp[n 2Pi I / 8] / n, {n, k}];{Re[s], Im[s]}, {k, 25}], DiscretePlot[s = Sum[1 / (n ^ 2 + I n + 2I), {n, k}];{Re[s], Im[s]}, {k, 25}]}Parametric Sums (6)
Exponential or geometric series:
SumConvergence[x ^ n, n]SumConvergence[1 / x ^ n, n]Parameter region for convergence:
{With[{x = u + I v}, RegionPlot[Abs[x] < 1, {u, -2, 2}, {v, -2, 2}]], With[{x = u + I v}, RegionPlot[Abs[x] > 1, {u, -2, 2}, {v, -2, 2}]]}SumConvergence[x ^ n / n!, n]SumConvergence[(-3) ^ n x ^ (2 n) / n, n]
SumConvergence[(x / 2) ^ n, n] && SumConvergence[x ^ (-n), n]With[{x = u + I v}, RegionPlot[Abs[x] < 2 && Abs[x] > 1, {u, -2, 2}, {v, -2, 2}]]SumConvergence[1 / n ^ (x + 1), n] && SumConvergence[(x / 2) ^ n, n] && SumConvergence[1 / x ^ n, n]With[{x = u + I v}, RegionPlot[Re[x] > 0 && Abs[x] < 2 && Abs[x] > 1, {u, -2, 2}, {v, -2, 2}]]SumConvergence[Piecewise[{{a ^ n, n ≥ 0}, {b ^ n, n < 0}}]z ^ n, n]RegionPlot[Abs[a z] < 1, {a, -2, 2}, {z, -2, 2}]Assuming z=u+ v to be complex:
With[{z = u + I v}, RegionPlot3D[Abs[a z] < 1, {a, -2, 2}, {u, -2, 2}, {v, -2, 2}, AxesLabel -> {a, u, v}]]SumConvergence[a ^ n b ^ m, {n, m}]RegionPlot[Abs[a] < 1 && Abs[b] < 1, {a, -2, 2}, {b, -2, 2}]Convergence on Intervals (2)
Test the convergence of 
on different intervals:
SumConvergence[1 / n ^ 3, {n, 1, ∞}]SumConvergence[1 / n ^ 3, {n, 0, ∞}]SumConvergence[1 / n ^ 3, {n, a, ∞}]Test the convergence of a parametric sum on different intervals:
SumConvergence[1 / (3n + a) ^ 2, {n, 1, ∞}]SumConvergence[1 / (3n + a) ^ 2, {n, -1, ∞}]SumConvergence[1 / (3n + a) ^ 2, {n, -∞, ∞}]Options (10)
Method (10)
Test the convergence of 
using the ratio test:
SumConvergence[a ^ n / n!, n, Method -> "RatioTest"]Test the convergence of 
using the ratio test:
SumConvergence[(2n + 1)! / ((5 n)! n), n, Method -> "RatioTest"]In this case the ratio test is inconclusive:
SumConvergence[1 / n, n, Method -> "RatioTest"]Test the convergence of 
using the root test:
SumConvergence[x ^ n / n, n, Method -> "RootTest"]Test the convergence of 
using the root test:
SumConvergence[((2n + 3/5n - 4))^n, n, Method -> "RootTest"]In this case the root test is inconclusive:
SumConvergence[(1 - 1 / n) ^ n, n, Method -> "RootTest"]The Raabe test works well for rational functions:
SumConvergence[n / (n ^ 3 + 2n + 1), n, Method -> "RaabeTest"]SumConvergence[1 / Sqrt[n], n, Method -> "RaabeTest"]SumConvergence[(Underoverscript[∏, k = 1, n](2 k - 1)/Underoverscript[∏, k = 1, n]2 k), n, Method -> "RaabeTest"]In this case the Raabe test is inconclusive:
SumConvergence[x ^ n, n, Method -> "RaabeTest"]Test the convergence of 
using the integral test:
SumConvergence[1 / Log[n] ^ 2, n, Method -> "IntegralTest"]Test the convergence of 
using the integral test:
SumConvergence[1 / (n Log[n] Log[Log[n]] ^ 2), n, Method -> "IntegralTest"]In this case the integral test is inconclusive:
SumConvergence[1 / Prime[n], n, Method -> "IntegralTest"]Applications (3)
Find the radius of convergence of a power series:
SumConvergence[x ^ n, n]Sum[x ^ n, {n, 0, Infinity}, GenerateConditions -> True]Find the interval of convergence for a real power series:
SumConvergence[(x ^ n) / (n 3 ^ n), n]As a real power series, this converges on the interval [-3,3):
SumConvergence[(x ^ n) / (n 3 ^ n), n, Assumptions -> x∈Reals]Prove convergence of Ramanujan's formula for 
:
SumConvergence[Sqrt[8] / 9801(4n)!(1103 + 26390n) / (n!) ^ 4 / 396 ^ (4n), n]Sqrt[8] / 9801Sum[(4n)!(1103 + 26390n) / (n!) ^ 4 / 396 ^ (4n), {n, 0, Infinity}]Properties & Relations (4)
Convergence properties are not affected by multiplication of constants:
{SumConvergence[2 1 / n, n], SumConvergence[1 / n, n]}Convergence is not affected by translating arguments:
{SumConvergence[1 / n ^ 2, n], SumConvergence[1 / (n - 5) ^ 2, n]}SumConvergence is automatically called by Sum:
SumConvergence[1 / n ^ 2, n]Sum[1 / n ^ 2, {n, ∞}]Many conditions generated by Sum are in effect convergence conditions:
Sum[x ^ n, {n, ∞}, GenerateConditions -> True]SumConvergence[x ^ n, n]With the setting VerifyConvergence->False, typically a regularized value is returned:
Sum[(-1) ^ n, {n, ∞}, VerifyConvergence -> False]SumConvergence is used in sum transforms such as ZTransform:
ZTransform[a ^ n, n, z, GenerateConditions -> True]SumConvergence[a ^ n z ^ (-n), n]GeneratingFunction[a ^ n, n, x, GenerateConditions -> True]SumConvergence[a ^ n x ^ n, n]ExponentialGeneratingFunction:
ExponentialGeneratingFunction[a ^ n n!, n, x, GenerateConditions -> True]SumConvergence[a ^ n n! x ^ n / n!, n]FourierSequenceTransform[a ^ n UnitStep[n], n, ω, GenerateConditions -> True]SumConvergence[a ^ n Exp[-I n ω], n]Neat Examples (1)
Conditionally convergent periodic sums:
f[n_, p_] := (Mod[n, p] - Mean[Mod[Range[0, p - 1], p]]) / Log[n]Table[SumConvergence[f[n, p], n], {p, 3, 30, 5}]Table[DiscretePlot[Sum[f[n, p], {n, 2, m}], {m, 2, 75}, PlotLabel -> (Mod[n, p] - Mean[Mod[Range[0, p - 1], p]]) / Log[n]], {p, 3, 30, 5}]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