Built-in Mathematica Symbol

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.

MORE INFORMATION

The following options can be given:

	Assumptions	$Assumptions	assumptions to make about parameters
	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

CLOSE ALL

Basic Examples (2)

Test for convergence of the sum

:

Test the convergence of

:

Find the condition for convergence of

:

Test for convergence of the sum

:

In[1]:=

Out[1]=

Test the convergence of

:

In[2]:=

Out[2]=

Find the condition for convergence of

:

In[1]:=

Out[1]=

Scope (14)

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:

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 (4)

The ratio test typically applies to exponential and hypergeometric terms:

In this case the ratio test is inconclusive:

The root test typically applies to exponential terms:

In this case the root test is inconclusive:

The Raabe test works well for rational functions:

In this case the Raabe test is inconclusive:

The integral test works well on logarithmic terms:

In this case the integral test is inconclusive:

Applications (1)

Find the radius of convergence of a power series:

Prove convergence of Ramanujan's formula for

:

Sum it:

Properties & Relations (5)

Convergence properties are not affected by multiplication of constants:

Convergence is not affected by translating arguments:

Sufficient conditions are typically given for convergence:

On the boundary of the convergence region the sum may converge or diverge:

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: