1. The Ratio Test: Let Section 10.5 – The Ratio and Root Tests be a positive series and

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

3. The Root Test: Let Section 10.5 – The Ratio and Root Tests be a positive series and 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. Alternating Series: A series whose terms are alternately positive and negative. Section 10.6 – Alternating Series: Absolute and Conditional Convergence The general forms are as follows:

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

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

8. 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. Section 10.6 – Alternating Series: Absolute and Conditional Convergence Ifconverges, thenconverges.(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: 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 convergent Example: Alternating Harmonic Series Harmonic Series converges conditionally