Series

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

• Section 11.2 – Series

Expected Educational Results

• Objective 19–01: I can explain the difference between a series and a sequence.

• Objective 19–02: I can classify a series as geometric, $p$$p$-series, harmonic, or telescoping.

• Objective 19–03: I can explain what it means for a series to converge and diverge.

• Objective 19–04: I can explain why ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\rightarrow\infty}{a_n}=0$ is a necessary but not sufficient condition for series convergence.

• Objective 19–05: I can use the appropriate Series Convergence Test to determine if a series converges or diverges.

• Objective 19–06: I can compute the interval on which a power series converges.

• Objective 19–07: I can compute the sum of a convergent geometric series.

• Objective 19–08: I can compute the sum of a telescoping series.

• Objective 19–09: I can use a geometric series to rewrite a repeating decimal as a rational number.

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.

Series

Activity 09

For each of the following series, use technology to

• evaluate the limit at infinity, and

• determine if the series converges or diverges.

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

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

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

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

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

6. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\sqrt{n+3}}{n^2+1}}$$\displaystyle\sum_{n=1}^{\infty}{\frac{\sqrt{n+3}}{n^2+1}}$

Use Technology to Find a Pattern to Convergence or Divergence of a Series

NOTE: The use of technology in this activity is to quickly identify patterns. On Assessments, you must show work on assessments to receive credit for all Calculus II work.

Mathematica

​x
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(* Example from Investigation 09-1 *)
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(* Limit at infinity *)
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Limit[(n+1)/(n^2+2), n->Infinity]
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(* Convergence or Divergence *)
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SumConvergence[(n+1)/(n^2+2), 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. Limit[ ] has two arguments:

   1. The general $n^{\text{th}}$$n^{\text{th}}$-term, $a_n$$a_n$;

   2. $n\text{->Infinity}$$n\text{->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. SumConvergence[ ] has two arguments:

   1. The general $n^{\text{th}}$$n^{\text{th}}$-term, $a_n$$a_n$;

   2. $n$$n$, the independent variable.

5. 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.

6. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator or to group $-1$$-1$ when raised to $n$$n$.

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

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

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

Investigation 10

From your results above, if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle \lim_{n\rightarrow\infty}{a_n}=0$, then what can be said about the convergence or divergence of the series? Explain.

Theorem: Necessary Condition for Series Convergence

If the series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{a}_n}$$\displaystyle\sum_{n=0}^{\infty}{a_n}$ is convergent, then a necessary condition for series convergence is ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\to\infty}{a_n}=0$.

NOTE: The converse of this theorem is NOT true, i.e., if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$$\displaystyle\lim_{n\to\infty}{a_n}=0$, then no conclusion can be made about the convergence or divergence of the series.

Theorem: Test for Divergence

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n\ne 0}$$\displaystyle\lim_{n\to\infty}{a_n}\ne 0$, then the series diverges.

Investigation 11

Determine if the following series converge or diverge.

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

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

3. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n+3}{\sqrt{n}+2}}$$\displaystyle\sum_{n=1}^{\infty}{\frac{n+3}{\sqrt{n}+2}}$

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

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

6. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n^2+1}{\sqrt{n+3}}}$$\displaystyle\sum_{n=1}^{\infty}{\frac{n^2+1}{\sqrt{n+3}}}$

Homework

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

• Section 11.2: # 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 57, 59.

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Created: Wednesday, 27 October 2021 - 06:55 (EDT)