Two related tests for checking series convergence

The ratio test and the root test are two more ways of checking for convergence of infinite series. The ratio test asks whether, in the limit that the number of terms goes to infinity (n → ∞), the ratio of the (n+1)th term to the nth term is less than one.

The root test checks whether the limit, as n → ∞, of the nth root of the nth term is less than one. We'll see why that implies series convergence in the proof.

These tests (particularly the ratio test) are valuable in the next phase of learning about infinite series, when we determine what range of a variable x allows the series to converge – its radius of convergence or its interval of convergence.

First we have to take a little digression to refine our understanding of convergence of a series.


Absolute vs. conditional convergence


We have dealt with alternating series, and the criteria for convergence of those are somewhat easier than for non-alternating (positive) series. Here we'll take a deeper look at the meaning of convergence. First consider the series

We've shown that this is a convergent p-series, with (p = 2) > 1. We can further claim that this series converges absolutely because

This is called the absolute convergence test. It's worth thinking about a counterexample to understand absolute convergence. Consider the series

Under the absolute convergence test, this series fails, because taking the absolute value of the term just gives us the divergent harmonic series with terms 1/n. Yet we know that this alternating series does converge. We say that it converges conditionally, the condition being that the terms alternate sign.

The test for conditional convergence is

That is certainly the case for the alternating harmonic series.

The last case is a divergent series like the simple harmonic series:

A series like this diverges by one of our established convergence tests (in this case the integral test works well), or its terms simply do not decrease, so it fails the divergence test.

Convergence refined

Test for absolute convergence

Test for conditional convergence

Divergence – the series diverges by any one of the tests for convergence, or the divergence test.

The ratio test

The ratio test is a pretty straightforward test for convergence, and can distinguish between absolute and conditional convergence for series with negative terms.

 

We'll state it here, then do some examples. Proofs of the tests and theorems on this page will be saved for last.

The ratio test

Then there are three possibilities for the limit, L:

Example 1

Ratio test, convergent series

Solution: We'll set up the ratio test for this series and see how it goes. In setting up these ratios remember that n has to become n+1 in the numerator:

Now generally we'll get a single fraction by recognizing that division is multiplication by the reciprocal:

Now we can use the properties of factorials and the laws of exponents to do some simplification. Recall that 2n+1 = 2n ยท 21 and (n + 2)! = (n + 2)(n + 1)!, so we have:

which reduces to

The limit is less than one, so the series converges. Hopefully, you had a pretty good suspicion that this series was convergent from the beginning, because the factorial function grows much more rapidly than an exponential function for n > N, some number at or beyond the point where the two functions cross.

Remember that this test, and any convergence test doesn't necessarily tell us which number the series converges to, just that it does. It doesn't give us the sum of the series, either. We would need to find that in another way.


Example 2

Ratio test, divergent series

Solution: First set up the ratio test:

Now, recognizing that

and using L'Hopital's rule to evaluate the limit, we get:

The series diverges. Notice that this series wouldn't have passed the divergence test, either.


Example 3

Ratio test, inconclusive

Solution: We begin by setting up the ratio-test expression. I'm going to forgo the absolute value signs this time (just assume they're there), and I'll just write the denominator as a reciprocal product right away:

Now expand the (n+1)2 and rearrange to keep the roots together:

The properties of limits allow us to separate this expression into a product of limits:

The properties of limits also allow us to put the power of 1/2 on the left outside of the limit ...

... to obtain a limit of 1, which means that the ratio test is inconclusive for this series.

This series can be shown to be convergent by direct comparison to the p-series with p = 3/2.


The root test


The root test is used in situations where a series term or part of it is raised to the power of the index variable. If the root test isn't fairly easy to use, you probably shouldn't use it. But when it works, it often cuts to convergence or divergence quickly.

The root test isn't a good choice if a series contains factorial terms. You'll notice some similarities between how we interpret the root test and the ratio test.

The root test

Then there are three possibilities for the limit, L:

Example 4

Root test, convergent series

Solution: This series is a perfect candidate for the root test because taking the nth root will vastly simplify it. Here goes:

Now we can use L'Hopital's rule to find the limit:

... so the series converges.

Series raised to a power of the index variable (the variable that counts in the summation) are good candidates for the root test.


Example 5

Root test, convergent series

Solution: Here is another great candidate for the root test because of the powers of n. Set up the limit like this:

Because 0 < 1, the series converges. Notice that the series with terms -3/n would not converge (-3 times a harmonic series, but that's not what's going on here).


Example 6

Root test, convergent series

Solution: First set up the limit. The nth root gives us 2/3.

Now the n3/n can be rewritten as (n1/n)3, which is one as n → ∞:

... so the series converges.


Proof of the ratio test


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