Direct comparison bounds an unknown series by a known one. Smaller than a convergent series converges; larger than a divergent one diverges.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The direction matters
Being smaller than a divergent series proves nothing. Only the two useful directions give a conclusion.
Limit comparison
Take the limit of the ratio of the two nth terms. A finite positive limit means both series do the same thing.
Why limit comparison helps
It avoids the inequality work. Comparing the dominant behaviour of the terms is usually enough.
The ratio test
Take the limit of the ratio of consecutive terms. Below 1 converges, above 1 diverges, and exactly 1 is inconclusive.
When to reach for it
The ratio test is built for factorials and exponentials, where consecutive terms simplify dramatically.
Direct comparison
If terms are smaller than those of a convergent series, the series converges. If larger than those of a divergent one, it diverges. The inequality must run the right way for the conclusion you want.
Limit comparison is more forgiving
If the ratio of the terms of two positive series tends to a finite positive limit, both converge or both diverge. It avoids constructing an inequality, which is often the awkward part of direct comparison.
The ratio test
Compute the limit of |aₙ₊₁/aₙ|. Less than 1 gives convergence, greater than 1 divergence, exactly 1 no information. It is the natural tool whenever factorials or nth powers appear.
The ratio test drives power series
Applying it to a power series and solving the resulting inequality gives the radius of convergence. That is the standard method, which is why the ratio test is worth particular fluency.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Sana Choose the Ratio Test
Sana examines the series whose nth term is 2ⁿ ÷ n factorial.
- Step 1
The term contains a factorial, which signals the ratio test.
Step 4: Your Turn
Practice makes it stick.
The Ratio
Problem 1 of 2
The ratio test gives a limit of 0.5. Does the series converge? 1 yes, 0 no.
The Inconclusive Case
Problem 2 of 2
The ratio test gives a limit of 1. Is the test conclusive? 1 yes, 0 no.
Pick a Test
1 of 8
Ratio test limit 0.3. Converge? 1 yes, 0 no.
2 of 8
Ratio test limit 2. Converge? 1 yes, 0 no.
3 of 8
Smaller than a convergent series. Does it converge? 1 yes, 0 no.
4 of 8
Smaller than a divergent series. Is anything proved? 1 yes, 0 no.
5 of 8
Limit comparison gives a finite positive limit. Do both series behave the same? 1 yes, 0 no.
6 of 8
A term contains a factorial. Which test first? 1 ratio, 2 comparison.
7 of 8
Match each test with when it is most useful.
Tap a card on the left to start.
8 of 8
Ratio test limit 0.9. Converge? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
The ratio test gives a limit of 1.5. Does the series converge? 1 yes, 0 no.
Question 2 of 2
When is the ratio test inconclusive?
What You Learned
- 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.