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.

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.

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.