Sequences

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

• Section 11.1 – Sequences

Expected Educational Results

• Objective 18–01: I can classify a sequence as arithmetic, geometric, or fibonacci.

• Objective 18–02: I can explain what it mean for a sequence to converge and diverge.

• Objective 18–03: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence converges or diverges.

• Objective 18–04: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence is increasing or decreasing.

• Objective 18–05: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence is bounded.

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.

Sequences

Convergence and Divergence of Sequences

Definition: Convergent Sequence

A sequence converges if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n}$$\displaystyle\lim_{n\to\infty}{a_n}$ exists.

Definition: Divergent Sequence

A sequence diverges if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n}$$\displaystyle\lim_{n\to\infty}{a_n}$ does not exist.

Activity 05

1. Let $a_n=\{1,1,1,1,1,1,...\}$$a_n=\{1, 1, 1, 1, 1, 1, ...\}$.

   1. Is $a_n$$a_n$ a geometric sequence? Explain.

   2. If $a_n$$a_n$ is a geometric sequence, then identify the common ratio, $r$$r$.

   3. Does $a_n$$a_n$ converge or diverge? Explain.

2. Let $a_n=\{1,-1,1,-1,1,-1,...\}$$a_n=\{1, -1, 1, -1, 1, -1, ...\}$.

   1. Is $a_n$$a_n$ a geometric sequence? Explain.

   2. If $a_n$$a_n$ is a geometric sequence, then identify the common ratio, $r$$r$.

   3. Does $a_n$$a_n$ converge or diverge? Explain.

NOTE: A geometric sequence converges when $-1<r\le 1$$-1< r \leq 1$.

Investigation 06

Determine if the following sequences converge or diverge. Explain.

1. ${\displaystyle a_n=\frac{n+1}{n}}$$\displaystyle a_n=\frac{n+1}{n}$, $n\ge 1$$n\geq 1$.

2. ${\displaystyle a_n=(-1{)}^n\frac{3n^2+1}{n+2}}$$\displaystyle a_n=(-1)^n\frac{3n^2+1}{n+2}$, $n\ge 0$$n\geq 0$.

3. ${\displaystyle a_n=n{e}^{-n}}$$\displaystyle a_n=ne^{-n}$, $n\ge 1$$n\geq 1$.

4. ${\displaystyle a_n=2{\left(\frac{2}{3}\right)}^n}$$\displaystyle a_n=2\left(\frac{2}{3}\right)^n$, $n\ge 0$$n\geq 0$.

5. ${\displaystyle a_n=\frac{1}{n}}$$\displaystyle a_n=\frac{1}{n}$, $n\ge 1$$n\geq 1$.

6. ${\displaystyle a_n=(-1{)}^{n+1}\frac{n}{n^2-3n+2}}$$\displaystyle a_n=(-1)^{n+1}\frac{n}{n^2-3n+2}$, $n\ge 3$$n\geq 3$.

7. ${\displaystyle a_n=\sqrt{n+1}}$$\displaystyle a_n=\sqrt{n+1}$, $n\ge 0$$n\geq 0$.

8. ${\displaystyle a_n=e^n}$$\displaystyle a_n=e^n$, $n\ge 0$$n\geq 0$.

Activity 07

Graphically verify your answers to Investigation 06 using the embedded DESMOS graph below:

Check Your Work

Use Technology to Verify Convergence or Divergence of Sequences

NOTE: The use of technology in this activity is to verify convergence or divergence of sequences. On Assessments, you must show work on assessments to receive credit for all Calculus II work.

Mathematica

​x
1
(* Example: a_n = (n+1)/(3+2n), n>=0 *)
2
(* Determine if a_n converges or diverges *)
3

4
Limit[(n+1)/(3+2n), 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 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. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator or to group $-1$$-1$ when raised to $n$$n$.

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

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

7. If Mathematica returns a real number, then the sequence diverges.

8. If Mathematica returns $\mathrm{\infty }$$\infty$ [recall: $\mathrm{\infty }\notin \mathbb{R}$$\infty\notin\mathbb{R}$], then the limit DNE and the sequence diverges.

Homework

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

• Section 11.1: # 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63.

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Created: Tuesday, 10 November 2020 6:38 EDT Last Modified: Wednesday, 27 October 2021 - 07:36 (EDT)