2. Warm Up 2. Consider the series: a)What is the sum of the series? b)How many terms are required in the partial sum to approximate the sum of the infinite series within 0.001?

3. Ratio and Root Tests for convergence/divergence

4. Let be a series with nonzero terms. Find the ratio: 1.If the limit is < 1, the series converges. 2.If the limit is > 1 (including infinity), the series diverges. 3.If the limit = 1, the test is inconclusive. Choose a different convergence test  The Ratio Test (Use for series with n n, n!, exponentials, etc.)

5. Example:

7. Determine the convergence or divergence of: 1. 2. 3. 4.

8. The root test (use if every thing is raised to the n th power) Let be a series with nonzero terms. Find: 1.If limit is < 1, converges. 2.If limit is > 1 (including infinity), diverges. 3.If limit = 1, inconclusive. Choose a different test.

9. Examples: 1. 2.

11. Guidelines for Convergence/Divergence. 1.Does the nth term approach 0? If not, the series diverges. 2.Is the series one of those special types – geometric, p-series, telescoping or alternating? 3.Can the integral test, root test, or ratio test be applied? 4.Can the series be compared directly or using a limit?

12. Putting it all together: Determine the convergence or divergence of the series below. If possible, find what the series converges to. 1. 2. 3. 4. 5. 6.