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The Integral Test; p -Series

The Integral Test; p -Series. Lesson 9.3. Divergence Test. Be careful not to confuse Sequence of general terms { a k } Sequence of partial sums { S k } We need the distinction for the divergence test If Then must diverge.

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The Integral Test; p -Series

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  1. The Integral Test; p-Series Lesson 9.3

  2. Divergence Test • Be careful not to confuse • Sequence of general terms { ak } • Sequence of partial sums { Sk } • We need the distinction for the divergence test • If • Then must diverge Note this only tells us about divergence. It says nothing about convergence

  3. Convergence Criterion • Given a series • If { Sk } is bounded above • Then the series converges • Otherwise it diverges • Note • Often difficult to apply • Not easy to determine { Sk } is bounded above

  4. The Integral Test • Given ak = f(k) • k = 1, 2, … • f is positive, continuous, decreasing for x ≥ 1 • Then either • both converge … or • both diverge

  5. Try It Out • Given • Does it converge or diverge? • Consider

  6. p-Series • Definition • A series of the form • Wherep is a positive constant p-Series test • Converges if p > 1 • Diverges if 0 ≤ p ≤ 1

  7. Try It Out • Given series • Use the p-series test to determine if it converges or diverges

  8. Assignment • Lesson 9.3 • Page 620 • Exercises 1 - 35odd

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