Cogito
AP Calculus BC · Chapter 4 · Lesson 2
Comparison and Ratio Tests
Judging a series against one you know.
12 problems · about 22 minutes · LIM-7.A
What this lesson teaches
The student applies the comparison, limit comparison and ratio tests.
- Direct comparison needs the useful direction to conclude anything.
- Limit comparison compares dominant behaviour and avoids inequalities.
- The ratio test suits factorials, and a limit of exactly 1 is inconclusive.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe ratio test gives a limit of 1.5. Does the series converge? 1 yes, 0 no.
Answer 0
Why No.
When is the ratio test inconclusive?
Answer When the limit is exactly 1.
Why A limit of exactly 1.
Ratio test limit 0.3. Converge? 1 yes, 0 no.
Answer 1
Why Below 1.
Ratio test limit 2. Converge? 1 yes, 0 no.
Answer 0
Why Above 1.
Smaller than a convergent series. Does it converge? 1 yes, 0 no.
Answer 1
Why Bounded above by something finite.
Build It Up
The same ideas with more to keep track of.
3 problemsSmaller than a divergent series. Is anything proved? 1 yes, 0 no.
Answer 0
Why The wrong direction.
Limit comparison gives a finite positive limit. Do both series behave the same? 1 yes, 0 no.
Answer 1
Why That is what the test concludes.
A term contains a factorial. Which test first? 1 ratio, 2 comparison.
Answer 1
Why Factorials cancel in a ratio.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each test with when it is most useful.
Answer Ratio test → Factorials and exponentials; Limit comparison → Terms behaving like a known p-series; nth term test → A quick first check for divergence
Why Each test has a signature situation.
Ratio test limit 0.9. Converge? 1 yes, 0 no.
Answer 1
Why Still below 1.
The Ratio: The ratio test gives a limit of 0.5. Does the series converge? 1 yes, 0 no.
Answer 1
Why Yes.
The Inconclusive Case: The ratio test gives a limit of 1. Is the test conclusive? 1 yes, 0 no.
Answer 0
Why No.