Ch 9.4 Radius of Convergence

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Ch 9.4 Radius of Convergence

• Calculus Graphical, Numerical, Algebraic by
• Finney, Demana, Waits, Kennedy

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Importance of Convergence
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Importance of Convergence
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Importance of Convergence
This equation is an identity only for x in
(-1,1). Convergence matters!
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What Youll Learn

• When does a power series converge?
• If a power series converges, over what interval
  of x?
• The nth Term Test for Divergence
• There are 4 tests for convergence
• The Comparison Test
• Absolute Convergence Test for Alternating Series
• The Ratio Test
• Endpoint Convergence

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This result is useless since the series is only
accurate at the center.
Note R is the radius of convergence.
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Cases of Convergence
Case 1 Converges over a radius of 1 or -1 lt x
lt 1.
Case 2 Converges for all values of x.
Case 3 Converges only at center or x a.
We will see an example for homework.
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The nth Term Test for Divergence for
non-negative terms only
First we need to rule out divergence
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Direct Comparison Test
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Example of Comparison Test
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Example of Comparison Test
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Note This test is for alternating series.
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Example of Absolute Convergence
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Example of Absolute Convergence
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If L 1, Ratio Test is Inconclusive
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Endpoint Convergence
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Endpoint Convergence
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Summing a Telescoping Series
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Summing a Telescoping Series