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Notes 9.4 – Sequences and Series

Notes 9.4 – Sequences and Series. I. Sequences. A.) A progression of numbers in a pattern. 1.) FINITE – A set number of terms 2.) INFINITE – Continues forever. B.) Notation - a k where a is the sequence and k refers to the k th term.

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Notes 9.4 – Sequences and Series

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  1. Notes 9.4 – Sequences and Series

  2. I. Sequences A.) A progression of numbers in a pattern. 1.) FINITE – A set number of terms 2.) INFINITE – Continues forever

  3. B.) Notation - akwhere a is the sequence and k refers to the kthterm. C.) Explicitly defined sequence– Allows us to substitute k into an equation to find the kthterm. D.) Recursively defined sequence– Defines each term by using the previous term.

  4. E.) Ex. 1 - Define the following sequence both explicitly and recursively.

  5. II. Types of Sequences A.) Arithmetic Sequence – any sequence with a common difference between terms. B.) Geometric Sequence - any sequence with a common ratio between terms.

  6. C.) Ex. 2- Define the following sequence both explicitly and recursively.

  7. D.) Constructing a Sequence – Ex. 3 - The third and sixth terms of a sequence are -12 and 48 respectively. Find an explicit formula for the sequence if it is 1.) arithmetic and 2.) geometric 1.) -

  8. 2.) Divide the two equations

  9. III. Fibonacci Sequence A.) Summation of successive terms. Can only be defined recursively B.) Ex. 4– ak = 1, 1, 2, 3, 5, 8, 13,…

  10. IV. Summation Notation A.) The way we express the sum of a sequence of n terms {a1, a2, a3, …, an}  “the sum of ak from k = 1 to n” k = the index of summation.

  11. B.) Ex. 5- C.) Ex. 6-

  12. V. Sum of a Finite Arith. Seq. A.)

  13. B.) Proof -

  14. C.) Ex. 7- Find the sum of the following sequences: A.) 3, 6, 9, 12,…, 21 B.) 111, 108, 105,…27

  15. VI. Sum of a Finite Geom. Seq. A.)

  16. B.) Ex. 8- Find the sum of the following sequence: 3, 6, 12, 24,…,48, 96 C.) Ex. 9- Find the sum of the following sequence: 10, 20, 30, 40,…

  17. D.) Ex. 9 - Find the sum of the following sequence: 0.1, 0.01, 0.001, 0.0001, …

  18. VII. Sum of a Infinite Geom. Seq. A.) A.K.A. an INFINITE SERIES B.) Not a true sum. This is considered a Partial Sum If it converges…

  19. If | r |< 1 C.) Ex. 10- Find the sum of the following series:

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