AP Calculus BC 9.4 Radius of Convergence Objective: able to use the n th – Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of numbers or the radius of convergence of a power series.
Do you think these series converge? Why or why not? (hint: write out a few terms) 1. ∑ 2. ∑ 3. ∑ 1
4. ∑ 5. ∑ 6. ∑
Convergence of a series means the total of the added terms comes to a finite value
The Convergence Theorem for Power Series
The number R is the radius of convergence , and the set of all values of x for which the series converges is the interval of convergence .
The nth – Term Test for Divergence
If the limit equals zero that does not mean that the series converges . It means that it could converge. If the limit equals zero, more tests have to be performed to prove whether or not the series converges. However, if the limit is not zero, we know the series diverges.
This test is used as a ______
7. ∑ 8. ∑ 9. ∑ 1
10. ∑ 11. ∑ 12. ∑
An inconclusive result means that you need to do more work.
The Direct Comparison Test
Similar to the comparison test for improper integrals, Section 8.4, p. 464. Hint: Geometric series are your friend, the trick is finding the right one
Does the series converge or diverge? 13. ∑ 14. ∑ 15. ∑ !
When your series fails the Divergence ( nth - Term) Test, you have to do MORE WORK!! The Direct Comparison test is nice, especially when you can find a friendly geometric series as your comparison series. The following test falls into the "More Work" category.
The Ratio Test
The Ratio Test is best used with exponents of n and factorials.
Don’t forget about L’Hôpital’s Rule for evaluating indeterminate limits. (Section 8.2, p. 445)
16. Does the series ∑ converge?
17. Does the series ∑ converge? !
The ratio test is great, but series terms are not always positive. If the series Σ|an| of absolute values converges, then Σ an is said to converge absolutely . This leads to the following…
Absolute Convergence Implies Convergence
18. Does the series ∑ converge? !
19. For what values of x does the series converge? − + − ⋯ + −1 + ⋯ Plan: a) find radius of convergence, b) check left endpoint of interval for convergence, c) check right endpoint
Rate yourself on how well you understood this lesson. I don’t get I sort of I understand I understand I got it! it at all get it most of it but I it pretty well need more practice 1 2 3 4 5 What do you still need to work on?