Absolute Convergence

In the previous pages, we considered positive series and showed that there are tools (Tests of Convergence) one may use to decide on the fate of the series. But, these tools are only valid for positive series and can not be used for any series. So one may ask the natural question: what do we do for general series? First, note that there exists a very natural way to generate a positive number from any given number. Indeed, if the number is already positive then clearly the job is already done. And, if the number is negative, one may just erase the negative sign in the front and generate a positive number. In other words, what we are talking about is the absolute value of the number. Let us summarize this:

Let tex2html_wrap_inline85 be a series of numbers (not necessarily positive). Consider the positive series tex2html_wrap_inline87 . What is the relationship (if any) between the convergence of tex2html_wrap_inline85 and the convergence of tex2html_wrap_inline87 ?

Example: Consider the series

displaymath93

The series of the absolute values gives

displaymath95.

This is a divergent series. Using the Alternating Series test, one may prove that the series

displaymath97

is convergent.

What this example shows is that the convergence of tex2html_wrap_inline99 and the convergence of tex2html_wrap_inline101 are not equivalent. So, we may still wonder what happened if the series tex2html_wrap_inline101 is convergent. In this case, the series tex2html_wrap_inline99 is convergent.

We have the following result:

1.
If the series tex2html_wrap_inline87 is convergent, then tex2html_wrap_inline85 is convergent. We will say that tex2html_wrap_inline85 is absolutely convergent.
2.
If the series tex2html_wrap_inline85 is convergent and the series tex2html_wrap_inline87 is divergent, we will say that tex2html_wrap_inline85 is conditionally convergent.

From the above example, we conclude that the series

displaymath97

is conditionally convergent.

Example: Check whether the series

displaymath121

is absolutely convergent.

Answer: Consider the series

displaymath123.

This is a positive series for which one may use any appropriate test. Clearly, we have

displaymath125.

Since the series tex2html_wrap_inline127 is convergent (by the p-test), then the Basic Comparison Test implies that the series

displaymath123

is convergent. Therefore, the series

displaymath121

is absolutely convergent.

Remark: It should be noticed here that directly proving that the series

displaymath121

is convergent is not an easy matter. You should try it.....

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

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