Theorem 5.6.1. (The Ratio Test, part I: Absolute Convergence).
- A.
- If the limit \(\ds \rho = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\) exists, and \(\rho < 1\text{,}\) then \(\sum a_n\) is absolutely convergent.
Section 5.6 Ratio and Root Tests
References.
- Calculus, Early Transcendentals by Stewart, Section 11.6.
The Ratio Test for Absolute Convergence.
Often the best way to show that a series converges is to show absolute convergence, by comparison to a geometric series with positive ratio \(r\text{.}\)
One way to do this is to note that for a geometric series, with \(a_n = ar^n\text{,}\) one has \(\ds\left|\frac{a_{n+1}}{a_n}\right| = |r|,\) and so check if a series behaves roughly like this for large \(n\text{:}\)
The basic idea is that for any \(r\) with \(\rho < r < 1\text{,}\) the convergent geometric series \(\sum r^n\) is “bigger”, so limit comparison gives convergence of \(\sum |a_n|\text{.}\)
Theorem 5.6.2. The Ratio Test, concluded: divergence and the possibility of no answer.
- B.
- If the above limit \(\rho\) exists but with \(\rho>1\text{,}\) the series diverges. (In fact it fails the \(n\)-th term test dramatically, because \(|a_n| \to \infty\text{.}\))
- C.
- If the limit \(\rho = 1\text{,}\) or the limit defining \(\rho\) does not exist, this test gives no answer. For example, all \(p\)-series have \(\rho = 1\text{,}\) but some converge while others diverge.
Checkpoint 5.6.3.
Test the series \(\ds \sum_{n=0}^\infty \frac{n^3}{3^n}\) for convergence.
Checkpoint 5.6.4.
Test the series \(\ds \sum_{n=0}^\infty \frac{n^n}{n!}\) for convergence.
Checkpoint 5.6.5.
Test the series \(\ds \sum_{n=0}^\infty \frac{r^n}{n!}\) for convergence.
Note that this is actually an infinite family of series, one for each choice of the value \(r\text{,}\) so you must answer for each value of \(r\text{.}\)
The \(n\)-th Root Test for Absolute Convergence.
A cousin of the Ratio Test compares to geometric series in a different way, using the fact that for a geometric series, \(\sqrt[n]{|a_n|} \to |r|\text{.}\)
Theorem 5.6.6. The Root Test.
- If the limit \(\ds \rho=\lim_{n \to \infty}\sqrt[n]{|a_n|}\) exists, and \(\rho < 1\text{,}\) then \(\sum a_n\) is absolutely convergent.
- If the above limit \(\rho\) exists but with \(\rho>1\text{,}\) the series diverges. (Again, \(|a_n| \to \infty\text{.}\))
- If the limit \(\rho=1\text{,}\) or the limit does not exist, again this test gives no answer. (Again, all \(p\)-series have \(\rho=1\text{,}\) but some converge while others diverge.)
Which of these two tests to use?
- The two tests work quite similarly; for example, if the Ratio Test works, then the Root Test also works, and gives the same value for \(\rho\text{.}\)
- The Ratio Test is often easier to use, so it is usually best to try it first.
- On the other hand, sometimes the Ratio Test fails but the Root Test succeeds, so it is good to have as a backup.
Study Guide.
Study Calculus Volume 2, Section 5.6 2 that relate to the Ratio Test; in particular
- Theorem 6 (stating the Ratio Test).
- Example 23.
- Checkpoint 21.
- Review the Problem-Solving Strategy, which summarizes ideas from the last several sections, and then do Exercise 25 and Checkpoint 23.
- Do one or several exercises from the ranges 317–327 and 364–367.
openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-testsopenstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests
