Root Test

Why did you need to learn about nth roots and algebra when you were in algebra class? It was so you could figure out when series converge, of course!

Root Test in Calculus

If you need to know if a series converges, but there is a power of \( n \) in it, then the Root Test is generally the go-to test. It can tell you if a series is absolutely convergent or divergent. This is different from most tests which tell you whether a series converges or diverges, but doesn't say anything about absolutely convergence.

One of the limits you will frequently need to apply the Root Test is

\[ \lim\limits_{n \to \infty} \frac{1}{\sqrt[n]{n}} = 1,\]

but why is that true. Showing that limit is actually equal to 1 uses the fact from properties of exponential functions and natural logs that

\[ e^{-\frac{\ln n}{n}} = \frac{1}{\sqrt[n]{n}}.\]

Since the exponential function is continuous,

\[ \begin{align} \lim\limits_{n \to \infty} e^{-\frac{\ln n}{n}} &= e^{-\lim\limits_{n \to \infty} \frac{\ln n}{n}} \\ &= e^{0} \\ &= 1, \end{align} \]

which gives you the desired result.

Root Test for Series

First, let's state the Root Test.

Root Test: Let

\[ \sum\limits_{n=1}^{\infty} a_n \]

be a series and define \( L \) by

\[ L = \lim\limits_{n \to \infty} \left| a_n \right|^{\frac{1}{n}}= \lim\limits_{n \to \infty} \sqrt[n]{\left| a_n \right|} .\]

Then the following hold:

1. If \( L < 1 \) then the series is absolutely convergent.

2. If \( L > 1 \) then the series diverges.

3. If \( L = 1 \) then the test is inconclusive.

Notice that, unlike many series tests, there is no requirement that the terms of the series be positive. However, it can be challenging to apply the Root Test unless there is a power of \( n \) in the terms of the series. In the next section, you will see that the Root Test is also not very helpful if the series is conditionally convergent.

Root Test and Conditional Convergence

Remember that if a series converges absolutely, then it is, in fact, convergent. So if the Root Test tells you that a series converges absolutely, then it also tells you that it converges. Unfortunately, it will not tell you if a conditionally convergent series actually converges.

In fact the Root Test often can't be used on conditionally convergent series. Take for example the conditionally convergent alternating harmonic series

\[ \sum\limits_{n \to \infty} \frac{(-1)^n}{n} .\]

If you try to apply the Root Test, you get

\[ \begin{align} L &= \lim\limits_{n \to \infty} \left| a_n \right|^{\frac{1}{n}} \\ &= \lim\limits_{n \to \infty} \left| \frac{(-1)^n}{n} \right|^{\frac{1}{n}} \\ &= \lim\limits_{n \to \infty} \left( \frac{1}{n} \right)^{\frac{1}{n}} \\ &= 1. \end{align} \]

So in fact the Root Test doesn't tell you anything about the series. Instead to tell that the alternating harmonic series converges you would need to use the Alternating Series Test. For more details on that test, see Alternating Series.

Root Test Rules

The most significant rule about the Root Test is that it doesn't tell you anything if \( L = 1 \). In the previous section, you saw an example of a series that converges conditionally, but the Root Test couldn't tell you that because \( L = 1 \). Next, let's look at two more examples where the Root Test isn't helpful because \( L = 1 \).

Root Test Examples

Let's look at a couple of examples where the Root Test is useful.