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

Increasing, Decreasing, and Monotonic Sequences

Definition: Increasing Sequence

A sequence increases if

• $a_{n+1}>a_n$$a_{n+1} > a_n$ for all $n\ge 0$$n\geq 0$, or

• $\frac{d}{dn}{a}_n>0$$\frac{d}{dn}a_n>0$ for all $n\ge 0$$n\geq 0$

Definition: Decreasing Sequence

A sequence decreases if

• $a_{n+1}<a_n$$a_{n+1} < a_n$ for all $n\ge 0$$n\geq 0$, or

• $\frac{d}{dn}{a}_n<0$$\frac{d}{dn}a_n<0$ for all $n\ge 0$$n\geq 0$

Definition: Monotonic Sequence

A sequence is monotonic if the sequence is increasing or decreasing for all $n$$n$.

Investigation 08

Determine if the following sequences increasing, decreasing, or not monotonic. 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=\frac{3n^2+1}{n+2}}$$\displaystyle a_n=\frac{3n^2+1}{n+2}$, $n\ge 0$$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=\frac{n}{n^2-3n+2}}$$\displaystyle a_n=\frac{n}{n^2-3n+2}$, $n\ge 3$$n\geq 3$.

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

Activity 09

Graphically verify your answers to Investigation 08 using the embedded DESMOS graph below:

Check Your Work

Use Technology to Verify if Sequences are Increasing or Decreasing

NOTE: The use of technology in this activity is to verify intervals of increase or decrease 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 increases or decreases *)
3

4
D[(n+1)/(3+2n), n]
5

6
(* To determine the interval on which a_n increases, *)
7
(* i.e., when the derivative is positive and n>=0 *)
8

9
Reduce[{D[(n + 1)/(3 + 2 n), n] >= 0, n >= 0}, n]
10

11
(* To determine the interval on which a_n decreases, *)
12
(* i.e., when the derivative is negative and n>=0 *)
13

14
Reduce[{D[(n + 1)/(3 + 2 n), 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 two arguments:

   1. The derivative of the general $n^{\text{th}}$$n^{\text{th}}$-term, $a_n$$a_n$ and set to $\text{<=0}$$\text{<=0}$ [for decreasing] You can use $\text{>=0}$$\text{>=0}$ to test for increasing sequences.

   2. The independent variable, $n$$n$.

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. 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. Since Mathematica returns FALSE for the last line of code, then the sequence never decreases.

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, 10 November 2020 6:38 EDT Last Modified: Wednesday, 27 October 2021 - 07:46 (EDT)