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Section 9.1 – Sequences

Section 9.1 – Sequences

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Section 9.1 – Sequences. Sequence. A sequence is a list of numbers written in an explicit order. First Term. n th Term. Second Term. Generally, we will concentrate on infinite sequences , that is, sequences with domains that are infinite subsets of the positive integers. - PowerPoint PPT Presentation

Text of Section 9.1 – Sequences

Section 9.1 Sequences

Section 9.1 SequencesSequenceFirst TermSecond Termnth TermGenerally, we will concentrate on infinite sequences, that is, sequences with domains that are infinite subsets of the positive integers.Recursive FormulaA formula that requires the previous term(s) in order to find the value of the next term.

Example: Find a Recursive Formula for the sequence below.2, 4, 8, 16,

Explicit FormulaA formula that requires the number of the term in order to find the value of the next term.

Example: Find an Explicit Formula for the sequence below.2, 4, 8, 16,

The Explicit Formula is also known as the General or nth Term equation.A sequence which has a constant difference between terms. The rule is linear.

Example: 1, 4, 7, 10, 13,

(generator is +3)Arithmetic Sequencesna(n)112437410513+3+3+3+3Discrete

Explicit FormulaRecursive FormulaWrite an equation for the nth term of the sequence:a(0) is not in the sequence! Do not include it in tables or graphs!White Board Challenge36, 32, 28, 24, n=1 n=2 n=3 n=440, n=0

4Sequences typically start with n=1First find the generatorThen find the n=0 term.

A sequence which has a constant ratio between terms. The rule is exponential. Example: 4, 8, 16, 32, 64,

(generator is x2)Geometric Sequencesnt(n)1428316432564x2x2x2x2Discrete0 1 2 3 4 5 6

Explicit FormulaRecursive FormulaWrite an equation for the nth term of the sequence:White Board Challenge n=1 n=2 n=3 n=4

n=0

3, 15, 75, 375, x5a(0) is not in the sequence! Do not include it in tables or graphs!Sequences typically start with n=1First find the generatorThen find the n=0 term. New Sequences n=1 n=2 n=3 n=4 n=5

White Board Challenge n=1 n=2 n=3 n=4 n=5

Monotonic Sequence

Example 2IF:

THEN:

Since the denominator is smaller:

OR

Bounded SequenceExampleLimit of a SequenceReminder: Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:Constant FunctionLimit of xLimit of a Power of xScalar Multiple

Reminder: Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:Sum/DifferenceProductQuotient Power

Example

Converges to 1

Diverges

Converges to 0White Board Challenge

Converges to

Absolute Value TheoremExample

Because of the Absolute Value Theorem, Converges to 0

Since the limit does not equal 0, we can not apply the Absolute Value Theorem. Itdoes not mean it diverges. Another test isneeded.The sequence diverges since it does not have a limit: -1,1,-1,1,-1,Theorem: Bounded, Monotonic Sequences

The sequence appears to be monotonic: It is increasing.The limit of the sequence appears to be 6.Since the sequence appears to be monotonic and bounded, it appears to converge to 6.