This is the last test for series convergence that we’re going to be looking at. As with
the Ratio Test this test will also tell whether a series is absolutely convergent or not
rather than simple convergence.
Root Test
Suppose that we have the series
. Define,
Then,
1.
if
the series is absolutely convergent (and hence convergent).
2.
if
the series is divergent.
3.
if
the series may be divergent, conditionally convergent, or absolutely
convergent.
A
proof
of this test is at the end of the section.
As with the ratio test, if we get
the root test will tell us nothing and
we’ll need to use another test to determine the convergence of the series. Also note
that if
in the Ratio Test then the Root Test will also give
.
We will also need the following fact in some of these problems.
Fact
Let’s take a look at a couple of examples.
Example 1
Determine if the following series is convergent or divergent.
Solution
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 Fall '08
 prellis
 Mathematical Series

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