1. MAT 1236 Calculus III Section 11.2 Series Part II http://myhome.spu.edu/lauw

2. HW… WebAssign 11.2 Part II Quiz: 11.2 II

3. Part II Introduce Geometric Series, Harmonic Series Test for Divergence

4. Standard Series #1 Geometric Series (G.S.)

5. If |r|<1, then If |r|  1, then is divergent

6. Proof:

11. PPFTNE State and prove the convergence of the geometric series.

12. Example 3

13. Please pay attention to the important details of the solutions Identify the series as G.S. with the parameters a, and r From the absolute value of r, conclude that the series is convergent or divergent (Determine the sum if it is required)

14. Example 4

15. Example 5 Find the value of x for which is convergent

16. Standard Series #2 Harmonic Series The harmonic series is divergent Note: It is not intuitively obvious that the harmonic series is divergent. Proof: (skip)

17. Theorem If is convergent then Why?

18. Theorem If is convergent then

19. Theorem If is convergent then

20. Theorem If is convergent then

21. Test For Divergence If, then is divergent

22. Example 6 By the Test for Divergence,…

23. Example 6

24. PPFTNE T or F? If, then is convergent

25. Theorem