1. 13. Section 10.3 Convergence Tests

2. Section 10.3 Convergence Tests EQ – What are other tests for convergence?

3. So far we know…. Geometric with r<1 converges Geometric with r>1 diverges Repeating decimal converges Telescoping series converges Harmonic Series diverges Nth term test for divergence (if, series diverges)

4. CONVERGENCE TESTS Advantage: Performed on the terms of the series (a n ), not on the partial sums (S N ). Disadvantages: 1.No universal test- “Not in my job description…” 2.Never wrong – but sometimes inconclusive-“I dunno…” 3.When conclusive, only determine whether the limit exists (or not) – not it’s value - “You want me to do all the work?”

5. 1) The n th Term Test (we looked at this yesterday) Unfortunately:

6. 2) Integral Test If f(x) is: Then: Disadvantages – not all a n are strictly decreasing either both converge or both diverge

7. Example Harmonic Series Positive, continuous and decreasing Diverges so series also diverges

8. Example Not always decreasing so can’t use test If you can’t use calculator to graph, use 1 st derivative test (must always be negative)

9. Example continued On calculator, find max Change series so past this value, then can use test X = 2.718 or e Diverges

10. Example Positive, continuous and decreasing Converges so series also converges BUT can’t say answer is 1 Series is only accumulating discrete values, integral accumulates all values

11. Question A) Converge B) Diverge

12. 3) p-Test Disadvantages – few series involve exactly 1/n p

13. Identify which series converge and which diverge.

14. 4) Direct Comparison Test Disadvantages

15. Example A) Converge B) Diverge

16. Example A) Converge B) Diverge

17. You come up with an example!!

18. 5) Limit Comparison Test Disadvantages

19. Example Compare to We know this one converges It doesn’t matter which way you divide Finite and positive so both converge

20. Example Compare to We know this one diverges Finite and positive so both diverge

21. Summary Nth term test – tests for divergence only Integral test – works if series is positive, continuous, and decreasing P-Series – Converges for p>1 and diverges for p≤1. Direct comparison test – find a known series that is always greater than or always less than to compare to - either both converge or both diverge Limit comparison test – find a similar known series to compare to – either both converge or both diverge

22. Assignment Pg. 581: #1-9 odd, 13, 15, 19-39 odd, 49-55 odd, 59-63 odd (2 days)