# 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.

# 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 , 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(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(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 , typically a regularized value is returned:

SumConvergence is used in sum transforms such as ZTransform:

## Neat Examples(1)

Conditionally convergent periodic sums: