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

Bounded Sequences

Definition: Bounded Above

A sequence is bounded above if there is a number $P$$P$ such that $a_n\le P$$a_n\leq P$ for all $n\ge 0$$n\geq 0$.

Definition: Bounded Below

A sequence is bounded below if there is a number $R$$R$ such that $a_n\ge R$$a_n\geq R$ for all $n\ge 0$$n\geq 0$.

Definition: Bounded Sequence

A sequence is bounded if the sequence is bounded above and bounded below.

Investigation 10

Determine if the following sequences bounded above, bounded below, and bounded. 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=(-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$.

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

Activity 11

Graphically verify your answers to Investigation 09 using the embedded DESMOS graph below:

Theorem: Monotonic Sequence Theorem

Every bounded, monotonic sequence is convergent.

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 - 07:49 (EDT)