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 –Friday’s material WILL be on the quiz. AP Calculus BC Wednesday, 06 April 2016

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 –Tomorrow’s (Friday’s) material WILL be on the quiz. AP Calculus BC Thursday, 07 April 2016

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

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

Series and Convergence

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

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 a n If this sequence of partial sums converges, then the series is said to converge.

Series and Convergence

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?

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

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

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 → ∞.

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

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

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

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

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

Series and Convergence

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.)

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

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 –Today’s material WILL be on the quiz.

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

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

TestSeriesConvergesDivergesComment 2)Geometric Series Series and Convergence Sum:

Series and Convergence

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

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