1. ALTERNATING SERIES series with positive terms series with some positive and some negative terms alternating series n-th term of the series are positive

2. ALTERNATING SERIES alternating series alternating harmonic series alternating geomtric series alternating p-series

3. THEOREM: (THE ALTERNATING SERIES TEST) ALTERNATING SERIES alternating decreasing lim = 0 convg Remark: The convergence tests that we have looked at so far apply only to series with positive terms. In this section and the next we learn how to deal with series whose terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign. Determine whether the series converges or diverges. Example:

4. ALTERNATING SERIES Determine whether the series converges or diverges. Example: THEOREM: (THE ALTERNATING SERIES TEST) alternating decreasing lim = 0 convg

5. ALTERNATING SERIES Determine whether the series converges or diverges. Example: THEOREM: (THE ALTERNATING SERIES TEST) alternating decreasing lim = 0 convg

6. THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) ALTERNATING SERIES satisfies the three conditions approximates the sum L of the series with an error whose absolute value is less than the absolute value of the first unused term the sum L lies between any two successive partial sums and the remainder, has the same sign as the first unused term. THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) satisfies the three conditions OR Example:

7. ALTERNATING SERIES Find the sum of the series correct to three decimal places. Example: THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) satisfies the three conditions OR

8. ALTERNATING SERIES Find the sum of the series correct to three decimal places. Example:

9. ALTERNATING SERIES The rule that the error is smaller than the first unused term is, in general, valid only for alternating series that satisfy the conditions of the Alternating Series Estimation Theorem. The rule does not apply to other types of series. REMARK: THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) satisfies the three conditions OR

10. ALTERNATING SERIES TERM-102

11. TERM-101 ALTERNATING SERIES

12. TERM-092 ALTERNATING SERIES

13. Is called Absolutely convergent DEF: convergent IF converges absolutely Test the series for absolute convergence. Example: Alternating Series, Absolute and Conditional Convergence

14. Is called Absolutely convergent DEF: convergent IF converges absolutely Test the series for absolute convergence. Example: Is called conditionally convergent DEF: Test the series for absolute convergence. Example: if it is convergent but not absolutely convergent. REM: convgdivg Alternating Series, Absolute and Conditional Convergence

15. Is called Absolutely convergent DEF: convergent IF converges absolutely Test the series for absolute convergence. Example: Is called conditionally convergent DEF: if it is convergent but not absolutely convergent. REM: convgdivg Alternating Series, Absolute and Conditional Convergence

16. Absolutely convergent THM: convergent Determine whether the series converges or diverges. Example: Alternating Series, Absolute and Conditional Convergence convg THM: convg

17. Absolutely convergent conditionally convergent convergent divergent Alternating Series, Absolute and Conditional Convergence

18. Is called Absolutely convergent DEF: convergent IF converges absolutely Choose one: absolutely convergent or conditionally convergent Example: Is called conditionally convergent DEF: if it is convergent but not absolutely convergent. REM: convgdivg Alternating Series, Absolute and Conditional Convergence

19. REARRANGEMENTS Divergent If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged. But this is not always the case for an infinite series. By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms. Alternating Series, Absolute and Conditional Convergence

20. REARRANGEMENTS Divergent convergent See page 719 Alternating Series, Absolute and Conditional Convergence

21. REARRANGEMENTS Absolutely convergent REMARK: with sum s any rearrangement has the same sum s Conditionally convergent Riemann proved that r is any real number there is a rearrangement that has a sum equal to r. Alternating Series, Absolute and Conditional Convergence

22. SUMMARY OF TESTS

23. Series Tests 1)Test for Divergence 2) Integral Test 3) Comparison Test 4) Limit Comparison Test 5) Ratio Test 6)Root Test 7)Alternating Series Test Special Series: 1)Geometric Series 2)Harmonic Series 3)Telescoping Series 4)p-series 5)Alternating p-series SUMMARY OF TESTS

24. 5-types 1) Determine whether convg or divg 2) Find the sum s 3) Estimate the sum s 4) How many terms are needed within error 5) Partial sums SUMMARY OF TESTS

35. TERM-101 ALTERNATING SERIES