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Calculus II (MAT 146) Dr. Day Wednesday April 2, 2014

Calculus II (MAT 146) Dr. Day Wednesday April 2, 2014. Sequences and Series: Searching for Patterns (11.1/11.2). Why Study Sequences and Series in Calc II?. Taylor Polynomials applet Re-Expression! Infinite Process Yet Finite Outcome . . . How Can That Be? Transition to Proof.

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Calculus II (MAT 146) Dr. Day Wednesday April 2, 2014

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  1. Calculus II (MAT 146)Dr. Day Wednesday April 2, 2014 • Sequences and Series: Searching for Patterns (11.1/11.2) MAT 146

  2. Why StudySequences and Seriesin Calc II? Taylor Polynomials applet • Re-Expression! • Infinite Process Yet Finite Outcome . . . How Can That Be? • Transition to Proof MAT 146

  3. Polynomial Approximators Our goal is to generate polynomial functions that can be used to approximate other functions near particular values of x. The polynomial we seek is of the following form: MAT 146

  4. MAT 146

  5. Polynomial Approximators Goal: Generate polynomial functionsto approximate other functions near particular values of x. Create a third-degree polynomial approximator for MAT 146

  6. Create a 3rd-degree polynomial approximator for MAT 146

  7. What is a Sequence? MAT 146

  8. List the first five terms of each sequence. MAT 146

  9. Some Sequence Calculations • If an = 2n−1, list the first three terms of the sequence. • The first five terms of a sequence bn are 8, 8.1, 8.11, 8.111, and 8.1111. Create a rule for the sequence, assuming this pattern continues. • For the sequence cn= (3n−2)/(n+3) : i) List the first four terms. ii) Are the terms of cngetting larger? Getting smaller? Explain. iii) As n grows large, does cnhave a limit? If yes, what is it? If no, why not? • Repeat (3) for this sequence: • Give an example of L’Hôspital’s Rulein action. MAT 146

  10. Sequence Characteristics Convergence/Divergence: As we look at more and more terms in the sequence, do those terms have a limit? Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence. Boundedness: Are there values we can stipulate that describe the upper or lower limits of the sequence? MAT 146

  11. What is an Infinite Series? We start with a sequence {an}, ngoing from 1 to ∞, and define {si} as shown. The {si} are called partial sums. These partial sums themselves form a sequence. An infinite series is the summation of an infinite number of terms of the sequence {an}. MAT 146

  12. What is an Infinite Series? Our goal is to determine whether an infinite series converges or diverges. It must do one or the other. If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges. MAT 146

  13. Notable Series A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge? MAT 146

  14. Notable Series The harmonic series is the sum of all possible unit fractions. MAT 146

  15. Notable Series A telescoping sum can be compressed into just a few terms. MAT 146

  16. Fact or Fiction? MAT 146

  17. Assignments WebAssign • Ch 9 T/F Review due tonight! • Ch11.1 due next week Monday: 4/7/14 • Ch 11.2 due next week Wednesday: 4/9/14 • WA Quiz #8 (11.1 and 11.2) due Sunday 4/13/14 • Integral Applications Review due Sunday 4/13/14 Test #3Thursday 4/3 and Friday 4/4 Help Session tonite! 6-7 pm STV 210 MAT 146

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