Sequences and Series Notes

9.1 Sequences and Series

Learning Targets

• Use sequence and function notations to write the terms of sequences.

Sequences

• Sequences can start with a subscript of 0 instead of 1, meaning 0 is in the domain of the sequence.

Sequences: Example 1

• Find the first 4 terms of the sequence $a_n = 3 + (-1)^n$

Sequences: Example 2

• Find the first 4 terms of the sequence $a_n = \frac{(-1)^n}{2n+1}$

Sequences: Defining a Unique Sequence

• Listing the first few terms is not sufficient to define a unique sequence; the nth term must be given.

Sequences: Finding the nth Term

• Write the expression for the apparent nth term ($a_n$) of each sequence:

  • a. 1, 3, 5, 7, …

  • b. 2, -5, 10, -17, …

Sequences: More nth Term Examples

• Write the expression for the apparent nth term ($a_n$) of each sequence:

  • a. 1, 5, 9, 13, …

  • b. 2, -4, 6, -8, …

Sequences: Recursive Definitions

• Some sequences are defined recursively, meaning terms are defined using previous terms.

• You need to be given a few first terms, and then can find the rest using the equation.

Sequences: Recursive Example

• Write the first 5 terms of the sequence defined recursively as:

  • $a_1 = 3$

  • an=2an−1+a1​ where n≥2

Sequences: Fibonacci Sequence

• The Fibonacci sequence is defined recursively as follows:

  • $a_0 = 1$

  • $a_1 = 1$

  • an=an−2+an−1​ where n≥2

• Write the first 6 terms.

Sequences: Fibonacci Numbers

• The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Factorials

• Many sequences involve terms defined using special products called factorials.

• What is a factorial?

Factorials: Definition

• Definition of Factorial: If n is a positive integer, then n factorial is defined as:

  • $n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot (n - 1) \cdot n$

  • As a special case, zero factorial is defined as $0! = 1$

Factorials: Examples

• $1! = 1$

• $2! = 1 * 2 = 2$

• $3! = 1 * 2 * 3 = 6$

• $4! = 1 * 2 * 3 * 4 = 24$

• Factorials follow the same conventions for order of operations as exponents, so $2! = 2(!)$ = 2(1∗2∗3…)

• Whereas $(2!)! = 1 * 2 * 3 * 4 * … * 2$

Factorials: Sequence Example

• Find the first five terms of the sequence given by $a_n = \frac{2^n}{n!}$ beginning with n = 0.

Factorials: Simplification

• Simplify each factorial:

  • a. $\frac{8!}{2!6!}$

  • b. $\frac{4!}{(n+1)!}$

  • c. $\frac{n!}{(n-1)!}$

Homework

• 9. 1 Homework - Webassign Due: Tuesday 4/22

• Please show ALL of your work on paper and turn it in on Teams when you are done.