Ratio test
Find the limit L of |a_(n+1) / a_n|. If L < 1 the series converges, if L > 1 it diverges, and if L = 1 the test says nothing.
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Analogy
Checking whether each step of a staircase is shorter than the last by a steady factor.
Example
a_(n+1) / a_n = 2 / (n + 1).As n grows, 2/(n+1) tends to 0.L = 0, less than 1.So the series converges.
In practice
Works very well with factorials and powers, such as n! or 2^n.
Practice
Flashcards
- What does the ratio test conclude when L < 1?The series converges.
Interview questions
- Does the series of 2^n / n! converge? Show your test.
- Use the ratio test
- a_(n+1)/a_n = 2/(n+1)
- Limit as n grows is 0
- 0 is less than 1, so it converges
Questions people ask
What if the ratio test gives L = 1?
The test is inconclusive. Both the harmonic series (which diverges) and the series of 1 over n squared (which converges) give a limit of 1. You need a different test, such as comparison or the integral test, to decide. The ratio test cannot separate these two cases.
Which series suit the ratio test best?
Those with factorials, such as 2 to the n over n factorial, or with exponentials like n over 3 to the n. The ratio of consecutive terms simplifies nicely, because most factors cancel and leave a simple fraction in n. For 2 to the n over n factorial the ratio is just 2 over n plus 1.