Integral Test

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

• Section 11.3 – The Integral Test and Estimation of Sums

Expected Educational Results

• Objective 21–01: I can analytically (i.e., not graphically and not numerically) verify all conditions needed to use the Integral Test for a given series.

• Objective 21–02: I can use the Integral 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.

Integral Test

Definition: Integral Test

Let $f(n)=a_n$$f(n)=a_n$. Suppose for some number $c$$c$,

• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle \lim_{n\rightarrow\infty}{a_n=0}$ [a necessary condition for series convergence];

• $f$$f$ is positive on $[c,\mathrm{\infty })$$[c,\infty)$;

• $f$$f$ is continuous on $[c,\mathrm{\infty })$$[c,\infty)$; and

• $f$$f$ is decreasing on $[c,\mathrm{\infty })$$[c,\infty)$;

then the series

• ${\displaystyle \sum _{n=c}^{\mathrm{\infty }}{a}_n}$$\displaystyle \sum_{n=c}^{\infty}{a_n}$ is convergent if and only if ${\displaystyle {\int }_c^{\mathrm{\infty }}f(n)dn}$$\displaystyle \int_{c}^{\infty}{f(n)}\,dn$ is convergent; and

• ${\displaystyle \sum _{n=c}^{\mathrm{\infty }}{a}_n}$$\displaystyle \sum_{n=c}^{\infty}{a_n}$ is divergent if and only if ${\displaystyle {\int }_c^{\mathrm{\infty }}f(n)dn}$$\displaystyle \int_{c}^{\infty}{f(n)}\,dn$ is divergent.

Explanation of Integral Test

The Integral Test is written as a conditional statement.

1. Although not technically a condition for the Integral Test, you should always start with the Test for Divergence, i.e., verify ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle \lim_{n\rightarrow\infty}{a_n=0}$.

2. There are three parts to the hypothesis of the conditional statement. You need to verify that all three (3) parts of the hypothesis are true:

   1. $f$$f$ is positive on $[c,\mathrm{\infty })$$[c,\infty)$;

      • You need to state how you verified this condition is true

   2. $f$$f$ is continuous on $[c,\mathrm{\infty })$$[c,\infty)$;

      • For normally-behaving functions used in this course, $f$$f$ is continuous on its domain

   3. $f$$f$ is decreasing on $[c,\mathrm{\infty })$$[c,\infty)$;

      • You need to show $\frac{d}{dn}f(n)<0$$\frac{d}{dn}f(n)<0$ starting at some value $n=c$$n=c$

3. There are two (2) conclusions to the conditional statement:

   1. “${\displaystyle \sum _{n=c}^{\mathrm{\infty }}{a}_n}$$\displaystyle \sum_{n=c}^{\infty}{a_n}$ is convergent if and only if ${\displaystyle {\int }_c^{\mathrm{\infty }}f(n)dn}$$\displaystyle \int_{c}^{\infty}{f(n)}\,dn$ is convergent”

   2. “${\displaystyle \sum _{n=c}^{\mathrm{\infty }}{a}_n}$$\displaystyle \sum_{n=c}^{\infty}{a_n}$ is divergent if and only if ${\displaystyle {\int }_c^{\mathrm{\infty }}f(n)dn}$$\displaystyle \int_{c}^{\infty}{f(n)}\,dn$ is divergent”

NOTE: You must show all work when evaluating the improper integral.

NOTE: A key question to keep in mind prior to using the Integral Test: “Can I find an antiderivative of $f(n)$$f(n)$?”

Use Technology to Analytically Determine Domain of $f(n)$$f(n)$

NOTE: On all Assessments, you may use technology to analytically find the domain of functions. You must show all other Calculus II related work.

Mathematica

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(* Example from Investigation 01-1 *)
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(* Find domain of ln(n)/n *)
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FunctionDomain[Log[x]/x, x]

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. FunctionDomain[ ] has at least two arguments:

   1. The function, in terms of $x$$x$;

   2. $x$$x$ where $x$$x$ is the independent variable.

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

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

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

   3. Click on local

   4. Read how to use the FunctionDomain[ ] 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. If Mathematica returns $\vee$$\or$, this is the mathematical abbreviation of “or”.

Use Technology to Analytically Determine Interval(s) on which $f(n)$$f(n)$ is Decreasing

NOTE: On all Assessments, you may use technology to analytically find the interval(s) on which functions are decreasing. You must show all other Calculus II related work.

Mathematica

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(* Example from Investigation 01-1 *)
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(* Find the interval(s) where ln(n)/n is decreasing *)
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Reduce[D[Log[x]/x,x]<0, x]

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. The inequality, in terms of $x$$x$;

      1. The inequality includes the derivative of $f(x)$$f(x)$: $\text{D[Log[x]/x,x]}$$\text{D[Log[x]/x,x]}$ which has two arguments:

         1. The function, in terms of $x$$x$

         2. $x$$x$ where $x$$x$ is the independent variable.

   2. $x$$x$ where $x$$x$ is the independent variable.

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. If Mathematica returns $\vee$$\or$, this is the mathematical abbreviation of “or”.

Investigation 01

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

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

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

3. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n^3}{n^4+1}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{n^3}{n^4+1}}$

4. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^6+1}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{n^2}{n^6+1}}$

5. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}n{e}^{-n}}$$\displaystyle \sum_{n=1}^{\infty}{ne^{-n}}$

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 01-1 *)
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SumConvergence[Log[n]/n, 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 11.3: # 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 27.

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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 6:13 EDT Last Modified: Monday, 20 June 2022 - 12:46 (EDT)