The Root and Ratio Tests

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.

The Ratio Test:

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.

Remark: What do we mean when we say we do not have a definite conclusion? Simply that one may come up with series for which L=1 (in both tests) and they are convergent and other ones which are divergent.
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 ratio-test 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 Taylor-series 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.