Comparison and Ratio Tests

Judging a series against one you know.

12 problems · about 22 minutes · LIM-7.A

NameDate

Figure — use these to answer the problems

Comparison judges an unknown series against a curve like this whose behaviour is already settled.

y = 1/x + 0

Point on the curve2

Figure 1

Straightforward practice. Get the method working first.

5 problems

The ratio test gives a limit of 1.5. Does the series converge? 1 yes, 0 no.
Answer
2
When is the ratio test inconclusive?
- When the limit is exactly 1.
- When the limit is zero.
Ratio test limit 0.3. Converge? 1 yes, 0 no.
Answer
Ratio test limit 2. Converge? 1 yes, 0 no.
Answer
Smaller than a convergent series. Does it converge? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

Smaller than a divergent series. Is anything proved? 1 yes, 0 no.
Answer
Limit comparison gives a finite positive limit. Do both series behave the same? 1 yes, 0 no.
Answer
A term contains a factorial. Which test first? 1 ratio, 2 comparison.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Match each test with when it is most useful.
Draw a line from each item on the left to its match on the right.
- Ratio test
- Limit comparison
- nth term test
- Factorials and exponentials
- Terms behaving like a known p-series
- A quick first check for divergence
Ratio test limit 0.9. Converge? 1 yes, 0 no.
Answer
The Ratio
The ratio test gives a limit of 0.5. Does the series converge? 1 yes, 0 no.
Answer
The Inconclusive Case
The ratio test gives a limit of 1. Is the test conclusive? 1 yes, 0 no.
Answer