1. A list of numbers following a certain pattern { a n } = a 1, a 2, a 3, a 4, …, a n, … Pattern is determined by position or by what has come before 3, 6, 12, 24, 48, … Lecture 21 – Sequences 1

2. Find the first four terms and the 100 th term for the following: Defined by n(position) 2

3. An arithmetic sequence is the following: with a as the first term and d as the common difference. Arithmetic Sequence 3

4. A geometric sequence is the following: Geometric Sequence 4 with a as the first term and r as the common ratio.

5. Convergence We say the sequence “converges to L” or, if the sequence does not converge, we say the sequence “diverges”. A sequence that is monotonic and bounded converges. 5

6. Monotonic and Bounded Monotonic: sequence is non-decreasing (non-increasing) Bounded: there is a lower bound m and upper bound M such that 6 Monotonic & Bounded: Monotonic & not Bounded: Not Monotonic & Bounded: Not Monotonic & not Bounded:

7. Example 1 – Converge/Diverge? 7 Example 2 – Converge/Diverge?

8. Example 3 – Converge/Diverge? 8 Growth Rates of Sequences: q, p > 0 and b > 1 Lecture 22 – Sequences & Series

9. Example 4 – Converge/Diverge? 9

10. Partial Sums Adding the first n terms of a sequence, the n th partial sum: 10 Series – Infinite Sums If the sequence of partial sums converges, then the series converges.

11. Find the first 4 partial sums and then the n th partial sum for the sequence defined by: 11 Example 1

12. The partial sum for a geometric sequence looks like: 12 Geometric Series

13. Find the sum of the geometric series: Geometric Series – Examples 13 Lecture 23 – More Series

14. Find the sum of the geometric series: Geometric Series – More Examples 14

15. 15 Telescoping Series – Example 1

16. 16 Telescoping Series – Example 1 – continued

17. Telescoping Series – Example 2 17

18. 18