Convergence Tests

Source: Principles of Mathematical Analysis

Let be a sequence. We call the partial sums of the series . If converges to , then the series converges.

Cauchy Criterion

Info

converges if and only if for every , there is an integer such that if .

A sequence converges iff it is Cauchy. This follows by definition of a Cauchy sequence.

Comparison Test

Info

1. If for where is a fixed integer, converging implies converges
2. If for , diverging implies diverges.

(1) By Cauchy Criterion, for , there is where implies

(2) This follows by contraposition

Cauchy Condensation Test

Info

Suppose is monotone decreasing. Then converges if and only if the following series converges.

Define the following sequences, , . We have the following cases

Thus by Comparison Test (with some reindexing), and are either both bounded or unbounded.

Root Test

Info

Let and .

1. if , converges.
2. if , diverges
3. if , the test gives no information

If , pick and integer such that for . Then , but converges by geometric series and converges by Comparison Test.

If there is a sequence such that , and for infinitely many values of . Thus diverges.

Note diverges, while converges. Both of these have .

Ratio Test

Info

The series

1. Converges if
2. Diverges if for all , where is some fixed integer.

(1) There is such that for . Then , so . By Comparison Test this converges.

(2) Similar bounding can be performed to show the sequence diverges.