Convergence Tests

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Lecture 4

• Convergence Tests
• for Positive Series

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Lecture 4 Objectives

• Determine the convergence or divergence of
  positive series using
• The Integral Test
• The Comparison Test
• The Limit Comparison Test
• The Ratio Test
• The Root Test

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Recall The nth-Term Test for Divergence

• If ?n an converges, then limn an 0
• In other words, If limn an ? 0 (or does
  not exist), then ?n an diverges.
• Caution If limn an 0, then ?n
  an may or may not converge.

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Example Show that the seriesis divergent.
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Example

• Consider the following so-called Harmonic
  series

• Question Is this series convergent?

• Answer No, but this is not obvious.

• Caution The term sequence actually converges to
  0.

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Why is the harmonic series divergent?Reason 1

• The Harmonic series

can be shown (see Picture) to represent an area
that is ? the integral
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Picture
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The Integral Test
Reason
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Picture
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Example

• Show that the p-series
• converges if p gt 1,

and diverges if p ? 1.
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Why is the harmonic series divergent?Reason 2

• The Harmonic series

is ? the divergent series
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The Comparison Test
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Example

• Determine whether the following series are
  convergent or divergent.

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The Limit Comparison Test
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Example

• Determine whether the following series are
  convergent or divergent.

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Example continued
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The Ratio Test
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Example

• Determine whether the following series are
  convergent or divergent.

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The Root Test
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Example

• Determine whether the following series are
  convergent or divergent.

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Lecture 4 Objectives (revisited)

• Determine the convergence or divergence of
  positive series using
• The Integral Test
• The Comparison Test
• The Limit Comparison Test
• The Ratio Test
• The Root Test

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• Thank you for listening.
• Wafik