tg

tg
tgt

Wednesday, 27 March 2013

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 tex2html_wrap_inline157 be a positive series. Assume thatdisplaymath159.
Then we have the following:
1.
If tex2html_wrap_inline161 , then the series tex2html_wrap_inline157 is convergent or;
2.
If tex2html_wrap_inline165 , then the series tex2html_wrap_inline157 is divergent or;
3.
Iff tex2html_wrap_inline169 , then the series tex2html_wrap_inline157 may be convergent or it may be divergent. In other words, we do not have a definite conclusion.
The Ratio Test:

Let tex2html_wrap_inline157 be a positive series such that tex2html_wrap_inline175 for any tex2html_wrap_inline177 . Assume thatdisplaymath179.
Then we have the following:
1.
If tex2html_wrap_inline161 , then the series tex2html_wrap_inline157 is convergent or;
2.
If tex2html_wrap_inline165 , then the series tex2html_wrap_inline157 is divergent or;
3.
If tex2html_wrap_inline169 , then the series tex2html_wrap_inline157 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
displaymath195.

Answer: First, let us take care of the Ratio Test. We have
displaymath197,
which clearly implies
displaymath199.
Hence, we have L=1.
Next, we consider the Root Test. We have
displaymath203.
Since
displaymath205,
then we have
displaymath207.
Again, we have L=1.
But, we know that tex2html_wrap_inline211 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
displaymath215
where a > 0.
Answer: Because of the factorial, we will use the Ratio Test. We have
displaymath219.
Since
displaymath221,
then the series tex2html_wrap_inline223 is convergent.
Note that, in this case, we must have
displaymath225
for any a >0. Moreover, you may wonder what is the total sum? Using the Taylor-series one can show that
displaymath229.

Example:Discuss the convergence of
displaymath231.

Answer: Because of the factorial, we will use the Ratio Test. We have
displaymath233.
Since
displaymath235
then the series tex2html_wrap_inline237 is divergent.
Post a Comment