Do Now: Exploration 1 on p.492

1. Do Now: Exploration 1 on p.492 Consider and 1. Show that the Ratio Test yields for both series. For : For :

2. Do Now: Exploration 1 on p.492 Consider and 2. Use improper integrals to find the areas shaded in Figures 9.13a and 9.13b for . (a) (b)

3. Do Now: Exploration 1 on p.492 Consider and 3. Explain how Figure 9.14a shows that diverges, while Figure 9.14b shows that converges. Figure 9.14a shows that is greater than . Since the integral diverges, so must the series. Figure 9.14b shows that is less than . Since the integral converges, so must the series.

4. Do Now: Exploration 1 on p.492 Consider and 4. Explain how this proves the last part of the Ratio Test. These two examples prove that L = 1 can be true for either a divergent series or a convergent series. The Ratio Test itself is therefore inconclusive when L = 1.

5. Additional Convergence Tests Section 9.5a

6. The Integral Test Let be a sequence of positive terms. Suppose that , where f is a continuous, positive, decreasing function of x for all (N a positive integer). Then the series and the integral either both converge or both diverge. For the graphical proof, we will let N = 1 for the sake of simplicity, but the illustration can be shifted horizontally to any value of N without affecting the logic of the proof.

7. The Integral Test 1 n 1 n– 1 2 3 n+ 1 2 3 n (a) (b) (a) The sum provides an upper bound for (b) The sum provides a lower bound for

8. Applying the Integral Test Does converge? The integral test applies because the function is continuous, positive, and decreasing for x > 1. Check the integral: Since the integral converges, so must the series. (side note: they do not have to converge to the same value)

9. Harmonic Series and p-series Any series of the form (p a real constant) is called a p-series. The p-series test: 1. Use the Integral Test to prove that converges if p > 1. The series converges.

10. Harmonic Series and p-series Any series of the form (p a real constant) is called a p-series. The p-series test: 2. Use the Integral Test to prove that diverges if p < 1. If 0 < p < 1: The series diverges. (If , the series diverges by the nth-Term Test)

11. Harmonic Series and p-series Any series of the form (p a real constant) is called a p-series. The p-series test: 3. Use the Integral Test to prove that diverges if p = 1. The series diverges. This last divergent series is called the harmonic series.

12. The Limit Comparison Test (LCT) Suppose that and for all (N a positive integer). 1. If then and both converge or both diverge. 2. If and converges, then converges. 3. If and diverges, then diverges.

13. Using the LCT Determine whether the given series converge or diverge. For n large, behaves like Compare the given series to and try the LCT:

14. Using the LCT Determine whether the given series converge or diverge. Since the limit is positive and diverges, also diverges.

15. Using the LCT Determine whether the given series converge or diverge. For n large, behaves like Since Compare the given series to : converges (this is a geometric series with r = 1/2), the given series also converges.

16. Using the LCT Determine whether the given series converge or diverge. Compare to For n large, the series behaves like : Since converges by the p-Series Test, the given series also converges (by the LCT).

17. More Guided Practice Determine whether the series converges or diverges. There may be more than on correct way to determine convergence or divergence of a given series. The series diverges by the p-Series Test The series converges, since it is geometric with

18. More Guided Practice Determine whether the series converges or diverges. There may be more than on correct way to determine convergence or divergence of a given series. The Integral Test: Since the integral diverges, the series diverges as well.