1. Warm Up Finish your open notes quiz from yesterday. When you come in, I will return your quiz to you. Remember, you may use your 4 pages of notes and the cheat sheet, but nothing else. This is an INDIVIDUAL quiz. If you finished the quiz yesterday, you may have it back to look over it, or you can work on something for another class. (OR start memorizing those formulas )

2. CONVERGENT VS. DIVERGENT ARITHMETIC AND GEOMETRIC MEANS Do Sequences Converge or Diverge?

3. Infinite Series Earlier, we said that you could only find the sum of an infinite geometric series if… |r|<1 When |r|<1, the numbers in your sequences approached or got closer to zero.

4. Convergent Sequences When numbers in a sequence approach or get closer to a specific number as you keep listing them, they are said to converge. Convergent sequences look like this on a graph:

5. Divergent Sequences Divergent sequences do not approach one unique number. Their terms either continue to get bigger and bigger or smaller and smaller without approaching a stopping point. They look like this on a graph:

6. Convergent or Divergent? So how do we tell if a sequence is convergent or divergent? If it is geometric we can tell by looking at the “r” value. But if it is NOT geometric, we have to use another method.  Look at the graph OR  Generate the first few terms to see if they look like they are approaching one particular number.

7. Look at the r value to determine if the following geometric sequences are convergent or divergent. a 1 = 3, a n = a n-1 -2 a 1 = 4, a n = a n-1 0.5 a n = 5(-½) n-1 a n = ¼(3) n-1 Geometric: Convergent or Divergent?

8. Geometric: Convergent or Divergent? You Try

9. Given a Graph: Convergent or Divergent? Look at the right side of the graph. If the dots get close to horizontal, then the graph is convergent. Otherwise, it is divergent.

10. Given a Graph: Convergent or Divergent? You Try.

11. Given a Sequence Rule: Convergent or Divergent? If it is EXPLICIT: You can plug it into your calculator and look at the end behavior. Examples: a n = -3n + 12 a n = If it is RECURSIVE: You must generate the first 8 – 12 terms of the sequence to determine whether or not they approach a particular number. Examples: a 1 = 9, a n = a n-1 + 4 a 1 = 36, a n = -½a n-1 + 3

12. Given a Sequence Rule: Convergent or Divergent? You Try a n = 2n 2 – 4n + 2 a n = (-1) n a n = a 1 = 1, a n = a n-1 – 2 a 1 = -5, a n = a n-1 (-.5) + 6

13. Practice There are 12 index cards around the room. For each sequence, you must decide if it is convergent or divergent, and EXPLAIN how you figured it out.

14. Arithmetic Means Geometric Means Arithmetic Means—the terms that fall IN BETWEEN two terms in an arithmetic sequence. Example: 2, __,__,__,10 Geometric Means—the terms that fall IN BETWEEN two terms in a geometric sequence. Example: 3,__,__,24 If Extra Time…

15. Write an arithmetic sequence that has four arithmetic means between 4.3 and 12.8 First: find the common difference Then: Use d to determine the missing means. Calculating Means

16. Write a sequence that has two geometric means between 480 and -7.5 First: Find the common ratio. Then: Use r to find the missing means. Calculating Means

17. You try two! Write a sequence that has six arithmetic means between 4.3 and 12.8. Write a sequence that has 2 geometric means between -4 and 13.5

18. Homework Work on practice problems.