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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.