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:
.
Then we have the following:
The Ratio Test:
.
Then we have the following:
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.
For more examples, please click HERE.
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Mohamed A. Khamsi