You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.
Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.
For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.
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
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
There really isn’t much to these problems other than
computing the limit and then using the root test. Here is the limit for this problem.
So, by the Root Test this series is divergent.
|
|
Example 2 Determine
if the following series is convergent or divergent.
Solution
Again, there isn’t too much to this series.
Therefore, by the Root Test this series converges
absolutely and hence converges.
Note that we had to keep the absolute value bars on the
fraction until we’d taken the limit to get the sign correct.
|
|
Example 3 Determine
if the following series is convergent or divergent.
Solution
Here’s the limit for this series.
After using the fact from above we can see that the Root
Test tells us that this series is divergent.
|
Proof of Root Test
|
First note that
we can assume without loss of generality that the series will start at as we’ve done for all our series test
proofs. Also note that this proof is
very similiar to the proof of the Ratio Test.
Let’s start off
the proof here by assuming that and we’ll need to show that is absolutely convergent. To do this let’s first note that because there is some number r such that .
Now, recall
that,
and because we
also have chosen r such that there is some N such that if we will have,
Now the series
is a geometric
series and because we in fact know that it is a convergent
series. Also because by the Comparison test the series
is
convergent. However since,
we know that is also convergent since the first term on
the right is a finite sum of finite terms and hence finite. Therefore is absolutely convergent.
Next, we need
to assume that and we’ll need to show that is divergent. Recalling that,
and because we know that there must be some N such that if we will have,
However, if for all then we know that,
This in turn
means that,
Therefore, by
the Divergence Test is divergent.
Finally, we
need to assume that and show that we could get a series that has
any of the three possibilities. To do
this we just need a series for each case.
We’ll leave the details of checking to you but all three of the
following series have and each one exhibits one of the
possibilities.
|
