Sec. 9.4: Comparisons of Series

1. Sec. 9.4: Comparisons of Series

2. Sec. 9.4: Comparisons of Series The tests we have used so far involved either fairly simple series or series with special characteristics. Any deviation, no matter how slight, would render a test nonapplicable.

3. Sec. 9.4: Comparisons of Series is geometric, is not. is a p-series, is not. is easily integrated, is not.

4. Sec. 9.4: Comparisons of Series The next two tests allow you to compare a complicated series with a simpler series whose convergence or divergence is known.

5. Sec. 9.4: Comparisons of Series Since the convergence of a series is not dependent on the first several terms, you could modify the test to require only that 0 < an ≤ bn for all n greater than some integer N.

6. AP Calculus BCMonday, 31 March 2014 • OBJECTIVETSW (1) use the Direct Comparison Test (DCT) to determine whether a series converges or diverges, and (2) use the Limit Comparison Test (LCT) to determine whether a series converges or diverges. • CROSS-SECTION PROJECTS • Put on the table on the side with the purple rubrics sheet underneath (do not attach). • ASSIGNMENTS DUE • Sec. 9.1 wire basket • Sec. 9.2  black tray • Sec. 9.3 to the right of the black tray • QUIZ: Sec. 9.1 – 9.3 will be given after the lesson.

7. Sec. 9.4: Comparisons of Series 0 < an ≤ bn and 0 < bn ≤ an and converges diverges If a larger series converges, then the smaller series also converges. If a smaller series diverges, then the larger series also diverges.

8. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following. Include these in your answer for all n≥ 1

9. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following. for all n≥ 1 So no conclusion can be made using Try a different series.

10. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following. Term-by-term comparison:

11. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following. When a different N value is used, be sure to state it.

12. QUIZ: Sec. 9.1 – 9.3 • You may use a calculator.

13. AP Calculus BCTuesday, 01 April 2014 • OBJECTIVETSW (1) use the Direct Comparison Test (DCT) to determine whether a series converges or diverges, and (2) use the Limit Comparison Test (LCT) to determine whether a series converges or diverges.

14. Sec. 9.4: Comparisons of Series Informally, the Direct Comparisons Test says – (1) If the larger series Σbn converges, then the smaller series Σan must also converge. (2) If the smaller series Σan diverges, then the larger series Σbn must also diverge.

15. Sec. 9.4: Comparisons of Series Limit Comparison Test The next test, the Limit Comparison Test, should be used when a term-by-term comparison would be cumbersome to show.

16. Sec. 9.4: Comparisons of Series

17. Sec. 9.4: Comparisons of Series and and converges diverges

18. Sec. 9.4: Comparisons of Series Ex:Given Series Comparison Series

19. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following.

20. Sec. 9.4: Comparisons of Series Ex: Determine the convergence or divergence of the following.