Additional Convergence Tests

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.6 – Absolute Convergence and the Ratio and Root Tests

• Section 11.7 – Strategy for Testing Series

Expected Educational Results

• Objective 23–01: I can identify and explain when a series is conditionally convergent.

• Objective 23–02: I can correctly cancel factorial factors.

• Objective 23–03: I can use the Ratio Test to determine if a series converges or diverges.

• Objective 23–04: I can use the Root Test to determine if a series converges or diverges.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Absolute and Conditional Convergence

Definition: Absolute Convergence

A series $\sum a_n$$\sum{a_n}$ is absolutely convergent if the series $\sum |a_n|$$\sum{|\,a_n\,|}$ is convergent.

Definition: Conditional Convergence

A series $\sum a_n$$\sum{a_n}$ is conditionally convergent if the series $\sum a_n$$\sum{a_n}$ is convergent but the series $\sum |a_n|$$\sum{|\,a_n\,|}$ is divergent.

Theorem

If a series $\sum a_n$$\sum{a_n}$ is absolutely convergent, then the series is convergent.

Investigation 01

1. Determine if ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{n}}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^n\frac{1}{n}}$ is convergent or divergent. Explain.

2. Determine if ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}|(-1{)}^n\frac{1}{n}|}$$\displaystyle \sum_{n=1}^{\infty}{\left|(-1)^n\frac{1}{n}\right|}$ is convergent or divergent. Explain.

3. Determine if ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{n}}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^n\frac{1}{n}}$ is absolutely convergent or conditionally convergent. Explain.

Homework

At this time, you should be able to complete the following assignments:

• Section 11.6: # 3, 5.

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Created: Tuesday, 17 November 2020 6:42 EDT Last Modified: Tuesday, 30 May 2023 – 23:06 (EDT)