Alternating Series

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

• Section 11.5 – Alternating Series

Expected Educational Results

• Objective 22–01: I can analytically (i.e., not graphically and not numerically) verify all conditions needed to use the Alternating Series Test for a given series.

• Objective 22–02: I can use the Alternating Series Test to determine if a series converges or diverge.

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.

Alternating Series

Definition: Alternating Series

An Alternating Series is a series whose terms are alternatively positive and negative, i.e., ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}[(-1{)}^n{a}_n]}$$\displaystyle\sum_{n=0}^{\infty}{\left[(-1)^n a_n\right]}$ or ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}[(-1{)}^{n+1}{a}_n]}$$\displaystyle\sum_{n=0}^{\infty}{\left[(-1)^{n+1} a_n\right]}$

Definition: Alternating Series Test

If the Alternating Series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}[(-1{)}^n{a}_n]=a_0-a_1+a_2-a_3+\dots }$$\displaystyle\sum_{n=0}^{\infty}{\left[(-1)^n a_n\right]}=a_0-a_1+a_2-a_3+\dots$ for $a_n>0$$a_n>0$ satisfies

• $a_{n+1}\le a_n$$a_{n+1}\leq a_n$ or $\frac{d}{dn}{a}_n<0$$\frac{d}{dn}a_n<0$ for all $n$$n$; and

• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\to\infty}{a_n}=0$

then the series is convergent.

Explanation of Alternating Series Test

The Alternating Series Test is written as a conditional statement.

1. There are two (2) statements to verify in the hypothesis of the conditional statement:

   1. “$a_{n+1}\le a_n$$a_{n+1}\leq a_n$ or $\frac{d}{dn}{a}_n<0$$\frac{d}{dn}a_n<0$ for all $n$$n$”

      1. Mathematica may be used to verify the above statement – you must state that you verified this hypothesis statement

   2. "${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\to\infty}{a_n}=0$”

      1. You need to use Calculus I to evaluate this limit

2. “then the series is convergent” is called the conclusion of the conditional statement.

   1. The conclusion is True ONLY IF the both hypothesis statements are True.

NOTE: The Alternating Series Test, can NOT be used to conclude series divergence.

Investigation 01

Determine whether the following series are convergent or divergent. Explain.

1. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{n}}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^n\frac{1}{n}}$

2. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{2n+3}{4n-1}}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^n\frac{2n+3}{4n-1}}$

3. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^{n+1}\frac{n^2}{3n^3+2n^2+5n+1}}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^{n+1}\frac{n^2}{3n^3+2n^2+5n+1}}$

4. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^{n-1}\frac{3n}{\sqrt{n^3+1}}}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^{n-1}\frac{3n}{\sqrt{n^3+1}}}$

5. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{\ln (n)}{n^2}}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^{n}\frac{\ln{(n)}}{n^2}}$

6. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\cos \left(\frac{\pi }{n}\right)}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^{n}\cos{\left(\frac{\pi}{n}\right)}}$

7. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^{n+1}{n}^2{e}^{-n}}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^{n+1}n^2e^{-n}}$

Check Your Work

Use Technology to Determine if Terms are Increasing or Decreasing

NOTE: On all Assessments, you may use technology to determine if the terms of a series are increasing or decreasing without showing any work.

Mathematica

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(* Example from Investigation 01-3 *)
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(* Determine terms are decreasing *)
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(* To determine the interval on which a_n decreases, *)
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(* i.e., when the derivative is negative and n>=0 *)
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Reduce[{D[n^2/(3n^3+2n^2+5n+1), n] <= 0, n >= 0}, n]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Reduce[ ] has at least two arguments:

   1. Two (2) pieces of code grouped with $\text{{ }}$$\text{\{\ \}}$:

      1. The derivative of the $n^{\text{th}}$$n^{\text{th}}$-term which is set $\text{<= 0}$$\text{<= 0}$ to verify decreasing

         • Use $\text{>= 0}$$\text{>= 0}$ to verify increasing

      2. $\text{n >= 0}$$\text{n >= 0}$ where $0$$0$ is the first index (label) of the series

   2. $n$$n$ where $n$$n$ is the index variable of the series.

3. For help on using the Reduce[ ] function:

   1. In Mathematica, execute the code: $\text{?Reduce}$$\text{?Reduce}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Reduce[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

6. Mathematica will return “True” (series converges) or “False” (series diverges).

Use Technology to Verify Limits

NOTE: On all Assessments, you may use technology to verify limits. You must show all Calculus II related work.

Mathematica

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(* Example from Investigation 01-3 *)
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(* Verify limit at infinity is 0 *)
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Limit[n^2/(3n^3+2n^2+5n+1), n->Infinity]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Limit[ ] has at least two arguments:

   1. The $n^{\text{th}}$$n^{\text{th}}$-term of the series;

   2. $\text{n->Infinity}$$\text{n->Infinity}$.

3. For help on using the Limit[ ] function:

   1. In Mathematica, execute the code: $\text{?Limit}$$\text{?Limit}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Limit[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Use Technology to Verify Convergence or Divergence of a Series

NOTE: On all Assessments, you may use technology to verify your series convergence/divergence. You must show all Calculus II related work.

Mathematica

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(* Example from Investigation 01-3 *)
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(* Verify convergence *)
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SumConvergence[(-1)^(n+1) n^2/(3n^3+2n^2+5n+1), n]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. SumConvergence[ ] has at least two arguments:

   1. The $n^{\text{th}}$$n^{\text{th}}$-term of the series;

   2. $n$$n$ where $n$$n$ is the index variable of the series.

3. For help on using the SumConvergence[ ] function:

   1. In Mathematica, execute the code: $\text{?SumConvergence}$$\text{?SumConvergence}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the SumConvergence[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

6. Mathematica will return “True” (series converges) or “False” (series diverges).

Homework

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

• Section 11.5: # 3, 5, 7, 11, 13, 15, 17, 23, 25.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 17 November 2020 6:49 EDT Last Modified: Wednesday, 10 November 2021 - 05:46 (EDT)