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Calculus II (MAT 146) Dr. Day Fri day April 18, 2014. Tests for Series Convergence Comparison Tests (11.4 ) WA Quiz #9: D ue Sunday at midnight!. Polynomial Approximators.
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Calculus II (MAT 146)Dr. Day Friday April 18, 2014 • Tests for Series Convergence • Comparison Tests (11.4) • WA Quiz #9: Due Sunday at midnight! MAT 146
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
Notable Series The harmonic series is the sum of all possible unit fractions. MAT 146
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
Geometric Series (1) Create a geometric series to represent the repeating decimal fraction 0.45454545454545… and then determine the common fraction equivalent to this repeating decimal fraction. • (2) A geometric series has first term a1 = 1 and common ratio x, |x| < 1. • (a) Represent this using sigma notation and then show the first five terms of the series. • (b) Using your knowledge of convergent geometric series, determine a representation for this sum. MAT 146
Notable Series A telescoping sum can be compressed into just a few terms. MAT 146
Fact or Fiction? MAT 146
Series Convergence or Divergence We have defined sequences and series and have considered a few particular series—geometric series, harmonic series, telescoping sums—and explored the convergence/divergence of these series. Our first test related to this is • The Divergence Test also known as The nth-Term Test: If the sequence of terms that comprise an infinite series DO NOT have a limit of 0, then the series DOES NOT converge. The series diverges. We now turn our attention more fully to the fate of infinite series. What strategies do we have for determining whether an infinite series converges or diverges? MAT 146
Our First Series Convergence Test:The nth-Term Testalso calledThe Divergence Test MAT 146
Proofs MAT 146
Converge or Diverge? MAT 146
Converge or Diverge? (9) MAT 146
Converge or Diverge? (12) MAT 146
Assignments WebAssign • WA 11.3 due tonight! MAT 146