Post on 17-Jan-2016
Calculus II (MAT 146)Dr. Day Monday, Nov 2, 2015
Proving Series Convergence (Chapter 11)
Monday, Nov 2, 2015 MAT 146
Next Test: Thurs 11/19 & Fri 11/20
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Fact or Fiction?
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What is an Infinite Series?
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We start with a sequence {an}, n going 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}.
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Notable Series
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The harmonic series is the sum of all possible unit fractions.
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Notable Series
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A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge?
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Notable Series
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A telescoping sum can be compressed into just a few terms.
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Our first Series Convergence Test:
The nth-Term Testalso called
The Divergence Test
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Infinite Series Convergence
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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.
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Converge or Diverge?
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Converge or Diverge? (9)
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Converge or Diverge? (12)
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Absolute Convergenceand
Conditional Convergence
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Why Study
Sequences and Seriesin Calc II?
Taylor Polynomials applet
Infinite Process Yet Finite Outcome . . . How Can That Be?
Transition to Proof
Re-Expression!
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Polynomial Approximators
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On of our goals this chapter 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:
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Polynomial Approximators
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Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
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Create a 3rd-degree polynomial approximator for
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Each statement is FALSE! Identify one
word, symbol, or expression and correct it
to make each statement TRUE.
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