Hello! In this lesson we’ll add our last two tests to our series convergence/divergence toolbox.
Lesson 18: Ratio and Root Tests
Learning goals:
- Use the ratio test to determine absolute convergence of a series.
- Use the root test to determine absolute convergence of a series.
- Describe a strategy for testing the convergence of a given series.
Topic:
- Volume 2, Section 5.6 Ratio and Root Tests (link to textbook section)
WeBWorK:
- Series – Ratio and Root Tests
Motivating question
Recall from Lesson 14 (link here) that geometric series are nice. The convergence or divergence of a geometric series depends entirely on the common ratio $r$. In that lesson, we said that even if a series is not geometric, if it’s practically eventually almost geometric, then we’ll still be able to use similar tools to determine whether it converges or diverges.
So today’s motivating question is: what does it mean to be practically eventually almost geometric and how can we use that information to determine whether it converges?
Determine whether the series converges absolutely, converges conditionally, or diverges:
\[\sum_{n=1}^\infty \left(\frac{1}{3} \right)^n\]
This is a geometric series with common ratio $r= \frac{1}{3}$. Since $|r| < 1$, we know this series converges. In fact, since all terms are already positive, this means that the series converges absolutely.
Introduction
Both the ratio test and root tests involve some hypothesis that makes the notion practically eventually almost geometric precise. We know that if a series is actually geometric with common ratio $r$, then it converges if $|r| <$ and diverges if $|r| \geq 1$. For each of the ratio and root tests, a limit $\rho$ (the Greek letter “rho”) will play the role of $r$. (For the ratio test $\rho$ will be a limit of a ratio of individual terms; for the root test, $\rho$ will be the limit of a root of individual terms.) The series will converge if $\rho <1$ and diverge if $\rho >1$. The ratio and root tests are inconclusive if $\rho = 1$; in this case, we’ll have to use another test from our toolbox to determine if the series converges or diverges.
Ratio test
Theorem (Ratio test): Let $\sum_{n=1}^\infty a_n$ be a series with $a_n \neq 0$ for all $n$. Define $\rho$ as follows:
\[\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]
Then:
- if $\rho < 1$ then the series $\sum_{n=1}^\infty a_n$ converges absolutely,
- if $\rho >1$ then the series $\sum_{n=1}^\infty a_n$ diverges,
- if $\rho = 1$, then the ratio test is inconclusive (another test is needed).
Something to notice is that $\rho$ is a limit of positive terms, so $\rho$ itself is always greater than or equal to zero. (Your textbook uses the notation $\rho$ but the videos below use the notation $L$…$\rho$ and $L$ represent the same thing.)
A basic ratio test example
Before we see how to use the ratio test in the wild, let’s first see what it says about the geometric series from Warmup exercise 1. Since it’s faster to see that this is a geometric series and conclude that it converges based on the value of $r$, we probably wouldn’t apply the ratio test, if we had our choice. But seeing how the ratio test works in this example will help us understand the relationship between $r$ and $\rho$ and what it means for another series to be practically eventually almost geometric.
Example: Let $\sum_{n=1}^\infty \left(\frac{1}{3} \right)^n$. To use the ratio test, we need to determine $a_n$ and $a_{n+1}$. The general term $a_n =\left(\frac{1}{3} \right)^n$ so that means that $a_{n+1} = \left(\frac{1}{3} \right)^{n+1}$. Now we’re ready to determine the limit $\rho$:
\[ \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]
\[= \lim_{n \to \infty}\left| \frac{\left(\frac{1}{3} \right)^{n+1}}{\left(\frac{1}{3} \right)^n} \right|\]
\[=\lim_{n \to \infty} \left| \frac{1}{3} \right| \]
\[= \frac{1}{3}.\]
So in this example where the series was actually geometric, $\rho$ is just $r$! Since $\rho < 1$, the series converges absolutely by the ratio test.
Limits at infinity – refresher
In general, we’ll need to take limits which are not trivial, like the one in the basic example is. If you would like a review of limits at infinity from Calculus I, see Lesson 18 from the MAT 1475 course hub (link here).
More ratio test examples
Videos 1 and 2 below each show an example applying the ratio test to determine the convergence of a series.
Videos 3 and 4 below show one particular example twice, first using the limit comparison test (Video 3) and one using the ratio test (Video 4). The first video includes an introduction about why the other tests would not appropriate for this example.
Root test
Theorem (Root test): Let $\sum_{n=1}^\infty a_n$ be a series with $a_n \neq 0$ for all $n$. Define $\rho$ as follows:
\[\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|}\]
Then:
- if $\rho < 1$ then the series $\sum_{n=1}^\infty a_n$ converges absolutely,
- if $\rho >1$ then the series $\sum_{n=1}^\infty a_n$ diverges,
- if $\rho = 1$, then the ratio test is inconclusive (another test is needed).
Notice that the statement of the root test is almost identical to the statement of the ratio test; the only part that is different is the definition of $\rho$. Once again, $\rho$ will always be greater than or equal to zero.
A basic root test example
Before we get too involved with more complicated examples, let’s go back to the series from Warmup exercise 1 again. Remember that this series is actually geometric, so it doesn’t require the root test, but applying the root test to it will help us understand how the root test works.
Example: Let $\sum_{n=1}^\infty \left(\frac{1}{3} \right)^n$. To use the root test, we need to determine $a_n$, which is just the general term, so $a_n =\left(\frac{1}{3} \right)^n$. Now we’re ready to determine the limit $\rho$:
\[ \rho = \lim_{n \to \infty} \sqrt[n]{a_n}\]
\[= \lim_{n \to \infty} \sqrt[n]{\left| \left(\frac{1}{3} \right)^n \right|} \]
\[=\lim_{n \to \infty} \left| \frac{1}{3} \right| \]
\[= \frac{1}{3}.\]
So, again, in this example where the series was actually geometric, $\rho$ is just $r$! Since $\rho < 1$, the series converges absolutely by the root test.
More root test examples
Video 5 below provides an introduction to the root test, along with a few examples. Typically, the root test is a good tool to use when $a_n$ is an $n$-th power, but one of the examples in Video 5 shows how to apply the ratio test when $a_n$ isn’t quite an $n$-th power.
Don’t forget!
Don’t forget that if you’re applying the ratio or root test and your limit $\rho$ is 1, then the test is inconclusive and you’ll have to try another tool in your series convergence/divergence toolbox.
Summary
Phew! We’ve reached the end of our lessons on the tests in our series convergence/divergence toolbox! Just like when we were learning about the techniques in our integration technique toolbox (Lessons 3 to 9), not only do we have to know how to use the each individual tool, we also have to know when to use which tool. That is, we have to know how to use the tools and we have to know how to use the whole toolbox. Here is a flow chart that can help you choose the appropriate tool.
A great way to learn how to use the whole series convergence/divergence toolbox is to practice using the individual tools on a variety of different series. Often, many different tools can be used correctly to show the convergence or divergence of a series! But sometimes one tool will stick out as the easiest one. Try using what you think is an appropriate test for a given series and then try using what you think is an inappropriate test for the same series. What does it look like when you use an inappropriate tool? Can you force it to work or is it useless in your example?
- In Lessons 13 and 14 we were introduced to the theory and basic definitions of sequences and series. We saw that a series converges if its sequence of partial sums converges. We also saw:
- geometric series, and
- telescoping series.
- Then in Lessons 15 to 18 we saw:
- the divergence test,
- the integral test,
- p-series,
- the direct comparison test,
- the limit comparison test,
- the alternating series test,
- absolute versus conditional convergence,
- the ratio test, and
- the root test.
Not bad! You should now navigate over to the final exam review sheet (link here) and practice applying these tools to the series questions on it.
Additional Video Resources
Determine whether the series converges absolutely, converges conditionally, or diverges:
- \[\sum_{n=1}^\infty \frac{n!}{n^25^n}\]
- \[\sum_{n=1}^\infty \left(\frac{n+1}{2n+3}\right)^{3n}\]
