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

Example: Consider the series

The series of the absolute values gives

.

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

is convergent.

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

We have the following result:

1.

If the series is convergent, then is convergent. We will say that is absolutely convergent.

2.

If the series is convergent and the series is divergent, we will say that is conditionally convergent.

From the above example, we conclude that the series

is conditionally convergent.

Example: Check whether the series

is absolutely convergent.

Answer: Consider the series

.

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

.

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

is convergent. Therefore, the series

is absolutely convergent.

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

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

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