Here we will discuss the most popular tests of convergence known for series. They are to be used only on positive series.
The Root Test:
 Let be a positive series. Assume that.
Then we have the following:

 1.
 If , then the series is convergent or;
 2.
 If , then the series is divergent or;
 3.
 Iff , then the series may be convergent or it may be divergent. In other words, we do not have a definite conclusion.

 Let be a positive series such that for any . Assume that.
Then we have the following:

 1.
 If , then the series is convergent or;
 2.
 If , then the series is divergent or;
 3.
 If , then the series may be convergent or it may be divergent. In other words, we do not have a definite conclusion.

Example: Use the Ratio and Root Tests for the harmonic series
.
Answer: First, let us take care of the Ratio Test. We have
,
which clearly implies
.
Hence, we have L=1.
Next, we consider the Root Test. We have
.
Since
,
then we have
.
Again, we have L=1.
But, we know that is convergent, if and only if, p >1.
Remark: Note that the ratiotest is very appropriate and useful when the series has factorial terms.
Example: Discuss the convergence of
where a > 0.
Answer: Because of the factorial, we will use the Ratio Test. We have
.
Since
,
then the series is convergent.
Note that, in this case, we must have
for any a >0. Moreover, you may wonder what is the total sum? Using the Taylorseries one can show that
.
Example:Discuss the convergence of
.
Answer: Because of the factorial, we will use the Ratio Test. We have
.
Since
then the series is divergent.