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 that

displaymath159.

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 that

displaymath179.

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.

For more examples, please click HERE.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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