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The Ratio Test:

Section 10.5 – The Ratio and Root Tests. The Ratio Test:. Let. be a positive series and. If then the series converges. If then the series diverges. If then the test is inconclusive. The Ratio Test is a good test to use when the series contains sequences such as and .

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The Ratio Test:

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  1. Section 10.5 – The Ratio and Root Tests The Ratio Test: Let be a positive series and If then the series converges. If then the series diverges. If then the test is inconclusive. The Ratio Test is a good test to use when the series contains sequences such as and .

  2. Section 10.5 – The Ratio and Root Tests Ratio Test and the Geometric Series Geometric Series convergent series divergent series

  3. Section 10.5 – The Ratio and Root Tests The Root Test: Let be a positive series and If then the series converges. If then the series diverges. If then the test is inconclusive. The Root Test is a good test to use when the series contains sequences using exponential expressions.

  4. Section 10.5 – The Ratio and Root Tests

  5. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Alternating Series: A series whose terms are alternately positive and negative. The general forms are as follows:

  6. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Alternating Series Test If: 1) 1) 2) 2) 3) , 3) then the alternating series is convergent. Example: convergent. Alternating Harmonic Series

  7. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Alternating Series Test 3) , 2) 1) If: then the alternating series is convergent. Example: 3) 2) 1) convergent

  8. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Absolute Convergence If all terms of an infinite series are replaced by their absolute values and the resulting series converges, then the series is considered to have Absolute Convergence. If converges, then converges. (The converse is not true.) A series that is convergent but not absolutely convergent is Conditionally Convergent. All tests for convergence of positive series may be used in testing for absolute convergence.

  9. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Example: 2) 1) for 3) convergent

  10. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Test for Absolute Convergence Use the Ratio Test

  11. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Example: 3) 1) 2) convergent Harmonic Series divergent Alternating Harmonic Series converges conditionally

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