1. Section 9.2 – Series and Convergence

2. Goals of Chapter 9

3. Summation Notation A compact notation (often called sigma notation) for sums is the following: Upper Limit of Summation General Term Lower Limit of Summation Index of Summation

4. Examples Evaluate: i=1i=2 i=3 Series investigate the following:

5. Infinite Sum 1 1 What is the area of the square? Cut the square in half and label the area of one section. Cut the unlabeled area in half and label the area of one section. Continue the process… Sum all of the areas: The general term is… Since the infinite sum represents the area of the square…

6. Infinite Series

7. Connecting Series and Sequences Find the sum of…

8. Partial Sums of a Series

9. Convergent or Divergent Series

10. Examples Why do 1, 3, 4 and 6 Diverge? The limit of the general term does not equal 0.

11. The n-th Term Test When determining if a series converges, always use this test first!

12. The Converse of The n-th Term Test Consider the two famous sequences below:

13. The Converse of The n-th Term Test The Alternating Harmonic Series appears to converge to ~0.69. The Harmonic Series appears to diverge.

14. The n-th Term Test When determining if a series converges, always use this test first!

15. The Harmonic Series Diverges … … The Left Hand Riemann Sum is equal to the Sum of the Harmonic Series.

16. The Harmonic Series Diverges …

17. … We can find the value of the improper integral:

18. The Harmonic Series Diverges Part 2 Investigate the sum:

19. The Alternating Harmonic Series Converges Investigate and plot the sum: The sum is bounded by 0.5 and 1. Each Successive term in the sequence of partial sums is between the two previous terms in this sequence. The sum must be between any two successive terms. We will find the actual value of the sum soon.

20. Arithmetic and Geometric Series

21. Since Geometric Series occasionally converge, we will focus on them.

22. Definition of a Geometric Series

23. White Board Challenge

24. Finite Sum of a Geometric Series What happens to the sum as the value of n increases to infinity? Multiply by r. Subtract the two equations. Solve for the sum. Check with the previous example.

25. Infinite Sum of a Geometric Series Depends on the value of r.

26. Convergent Geometric Series Where a is the first term and r is the constant ratio.