1. OBJECTIVE TSW (1) list the terms of a sequence; (2) determine whether a sequence converges or diverges; (3) write a formula for the nth term of a sequence; and (4) use properties of monotonic sequences and bounded sequences. ASSIGNMENTS DUE MONDAY, 04/11/16 –Sequences –Series and Convergence Cross Sections projects due Friday ! ! ! QUIZ: Sequences, Series & Convergence, Integral & p-Series Tests will be on Monday, 11 April 2016. –Friday’s material WILL be on the quiz. AP Calculus BC Wednesday, 06 April 2016

2. OBJECTIVE TSW (1) list the terms of a sequence; (2) determine whether a sequence converges or diverges; (3) write a formula for the nth term of a sequence; and (4) use properties of monotonic sequences and bounded sequences. ASSIGNMENTS DUE MONDAY, 04/11/16 –Sequences –Series and Convergence Cross Sections projects due tomorrow ! ! ! QUIZ: Sequences, Series & Convergence, Integral & p-Series Tests will be on Monday, 11 April 2016. –Tomorrow’s (Friday’s) material WILL be on the quiz. AP Calculus BC Thursday, 07 April 2016

3. Series and Convergence (due on Monday, 04/11/16) Find the sum of the convergent series. Determine the convergence or divergence of the series.

4. Sequences Determine whether the following sequences are bounded and/or monotonic. bounded and monotonic monotonic and bounded below bounded but not monotonic

5. Series and Convergence

6. If {a n } represents a sequence, then the sum of the terms of a sequence represent an infinite series (or simply a series).

7. Series and Convergence To find this sum, consider the sequence of partial sums: S 1 = a 1 S 2 = S 1 + a 2 = a 1 + a 2 S 3 = S 2 + a 3 = a 1 + a 2 + a 3... S n = S n – 1 + a n = a 1 + a 2 + a 3 +... + a n If this sequence of partial sums converges, then the series is said to converge.

8. Series and Convergence

9. When dealing with series, there are two basic questions to consider: (1)Does a series converge or diverge? (2)If a series converges, what is its sum?

10. Series and Convergence Ex:Determine the convergence / divergence of the following series: List the partial sums: The Series converges and its sum = 1.

11. Series and Convergence Ex:Determine the convergence / divergence of the following series: The nth partial sum is

12. Series and Convergence This is an example of a telescoping series. It is of the form The nth partial sum is S n = b 1 – b n + 1 ; it follows that a telescoping series will converge if and only if b n approaches a finite number as n → ∞.

13. Series and Convergence Ex:Find the sum of the series Method #1: Partial Sums

14. Series and Convergence Ex:Find the sum of the series Method #2: Partial Fractions

15. Series and Convergence Ex:Find the sum of the series Method #2: Partial Fractions

16. TestSeriesConvergesDivergesComment 1) Telescoping Series Test Series and Convergence Sum: S = b 1 – L

17. Series and Convergence Geometric Series A geometric series is of the form with a common ratio r.

18. Series and Convergence

19. Ex:Determine the convergence / divergence of the following geometric series. If the series converges, find its sum. Identify r and a. State this. (Be sure to include absolute value.)

20. Series and Convergence Ex:Determine the convergence / divergence of the following geometric series. If it converges, find its sum.

21. AP Calculus BC Friday, 08 April 2016 OBJECTIVE TSW (1) understand the definition of a convergent infinite series; (2) Use properties of infinite geometric series; and (3) use the nth-Term Test for Divergence of an infinite series. ASSIGNMENTS DUE MONDAY, 04/11/16 –Sequences –Series and Convergence TODAY’S ASSIGNMENT (due on Tuesday, 04/12/16) –WS The Integral Test; p-Series Cross Sections projects due today ! ! ! QUIZ: Sequences, Series & Convergence, Integral & p-Series Tests will be on Monday, 11 April 2016. –Today’s material WILL be on the quiz.

22. Series and Convergence Ex:Use a geometric series to represent as the ratio of two integers. This can be written as a series: a

23. Series and Convergence Ex:Use a geometric series to represent as the ratio of two integers. This can be written as a series: a

24. TestSeriesConvergesDivergesComment 2)Geometric Series Series and Convergence Sum:

25. Series and Convergence

28. Ex:Apply the n th -Term Test to the following: The limit ≠ 0, so the series diverges. The limit = 0; the nth-Term Test fails. Another test must be used. Test: nTT

29. TestSeriesConvergesDivergesComment 3)nth-Term Test Series and Convergence This test cannot be used to show convergence