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Mathematical Expression Editor

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Some infinite series can be compared to geometric series.

We learned that the ratio test is a powerful technique based on the concept of
recognizing when a series is “approximately” geometric. If then the “tail” of the
series looks like a geometric series of ratio , and follows the same convergence and
divergence behavior as a geometric series when . The root test uses a similar idea in a
slightly different situation.

The Root Test If is an infinite series of positive terms,
and , then

if , then the series converges.

if or is infinite, then the series diverges.

if , the test is inconclusive. The series could diverge or converge.

Notice that the conclusion of the root test follows exactly the same form as the ratio
test. It does so for exactly the same reason:

If for large and , then for large , which says that the tail of behaves
like a geometric series with ratio .

Again, we do not give a formal proof in this course (but if you are interested, you can
find a proof online!)

When using the ratio test, there are two other subjects we should keep in mind. The
first subject is the rules of exponents. If you can’t remember these rules, pause here
and look them up to refresh your memory! The second subject is L’Hôpital’s rule,
which we used to evaluate limits like This limit, in particular, can appear
frequently!

Consider Discuss the convergence of this series.

We will attempt to use
the root test. Setting . Write with me. So, the root test says the series
is convergentsays the series is divergentgives no information in this
case, but we know the series is convergent using the integral testgives no
information in this case, but we know the series is divergent using the integral
test.

So the series is convergent by the root test.

Consider Discuss the convergence of this series.

We will attempt to use the
root test. Setting . Write with me. So, the root test says the series is
convergentsays the series is divergentgives no information in this case, but
we know the series is convergent through some other methodgives no
information in this case, but we know the series is divergent through some other
method.

So the series is divergent by the root test.

Consider Discuss the convergence of this series.

We will attempt to use the
root test. Setting . Write with me. So, the root test says the series is
convergentsays the series is divergentgives no information in this case, but
we know the series is convergent through some other methodgives no
information in this case, but we know the series is divergent through some other
method.

So the root test gives no information. However, we know this series diverges by the
divergence test.

Consider Discuss the convergence of this series.

We will attempt to use the
root test. Setting . Write with me. So, the root test says the series is
convergentsays the series is divergentgives no information in this case, but
we know the series is convergent through some other methodgives no
information in this case, but we know the series is divergent through some other
method.

So the root test gives no information. However, we know this series converges by the
-test.

When analyzing a series for convergence or divergence, choosing which test to use is
often the most difficult task we face. Generally, the root test is most useful when you
have a lot of powers and no factorials. Anytime you see a factorial is a pretty good
time to try the ratio test. Of course, don’t forget to use the divergence test
first!