AP Calculus BC 9.5 Testing Convergence at Endpoints Objective: able to determine the convergence or divergence of p-series; to use the Integral Test & the Alternating Series Test to determine the convergence or divergence of a series; to determine the absolute convergence, conditional convergence or divergence of a power series at the endpoints of its interval of convergence.
The Integral Test If f (x) is a positive, decreasing, and continuous function on [1, ∞) such that f (n) = an, then If converges, then ∑ converges.
If diverges, then ∑ diverges.
The interval [1, ∞) does not need to start at 1, and the function f (x) does not always have to be decreasing, just so long as it is decreasing for all values greater than or equal to the defined positive interval.
1. Does the series ∑ converge? (don’t forget the tests from the last Section 9.4))
2. Show that the harmonic series ∑ diverges. 3.What about ∑ ? .
The p-Series Test for Convergence For series of the form ∑ , If p ___ 1, the series converges If p ___ 1, the series diverges
The p-Series Test is a shortcut for applying the Integral Test
4. Does the series ∑ converge? √
The Limit Comparison Test Given a series ∑ (with an > 0 ) to test, and a known to converge/diverge series ∑ (with bn > 0 ), If lim = , 0 < < ∞ , then ∑ converges/diverges based on ∑ . →
If lim = 0 and ∑ converges, then ∑ converges. →
If lim = ∞ and ∑ diverges, then ∑ diverges. → To help remember the different cases, think of the following, if the value c > 0, then for n large, an ≈ c b n . The series ∑ and ∑ are roughly multiples of each other.
Useful when the given series resembles a known series that converges/diverges, but the direct comparison test is difficult to use.
5. Does the series ∑ converge?
The Alternating Series Test
6. Does the series ∑ converge absolutely, conditionally, or diverge?
Our purpose in this section has been to develop tests for convergence that can be used at the endpoints of the intervals of absolute convergence of power series. There are three possibilities at each endpoint: The series could diverge, it could converge absolutely, or it could converge conditionally.
7. For the series ∑ , find the a) interval of convergence, and values of x for which the series ! converges b) absolutely, c) conditionally.
8. For what values of x does the series ∑ converge?
9. For the series ∑ , find the a) interval of convergence, and values of x for which the series converges b) absolutely, c) conditionally.
Rate yourself on how well you understood this lesson. I have no clue what to do I can do it if someone is I can kind of do it on my I can do it on my own AND I even if somebody is walking me through the own, but I need the help of I can do it on my own can explain it to somebody explaining the problem to me problem my notes/textbook else 1 2 3 4 5 What do I still need to work on?