Understanding Alternating Series Convergence Tests and Theorems

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. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES TEST) 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. Example: Determine whether the series converges or diverges.

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

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

6. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) Example: THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) satisfies the three conditions 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 OR the remainder, has the same sign as the first unused term.

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

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

9. ALTERNATING SERIES THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM) REMARK: 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. satisfies the three conditions OR

10. ALTERNATING SERIES TERM-102

11. ALTERNATING SERIES TERM-101

12. ALTERNATING SERIES TERM-092

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

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

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

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

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

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

19. Alternating Series, Absolute and Conditional Convergence 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.

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

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

22. SUMMARY OF TESTS

23. SUMMARY OF TESTS Special Series: Series Tests • Geometric Series • Harmonic Series • Telescoping Series • p-series • Alternating p-series • Test for Divergence • Integral Test • Comparison Test • Limit Comparison Test • Ratio Test • Root Test • Alternating Series Test

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

35. ALTERNATING SERIES TERM-101