15.3 Criteria for Absolute Convergence

We assume in the following discussion that we are talking about the absolute values of the terms in your sequence, so they are all positive.

Then if each of your terms is smaller than the corresponding terms in a series you know to converge, your series will also converge; and conversely, if your terms are larger than those in a series that you know to diverge, then your series will diverge.

This statement is called "the comparison test" for absolute convergence.

But what series can I compare to?

The standard series to compare to is the geometric series.

This series has the wonderful property that if you multiply it by x and add 1 you have it back again. Thus, if you call it y, then it obeys xy +1 = y, which we can immediately solve for y. This simple fact means you must not and cannot forget the formula for this series, which is (1-x)-1.

It absolutely converges for |x| < 1, and diverges otherwise. By comparison to it, any other series will converge if the ratio of successive terms (later divided by earlier) is strictly less than some number that is less than 1. Your series will diverge if the ratio of successive terms is greater than 1.

This statement is called the ratio test for absolute convergence.

And what happens if the ratio of successive terms approaches 1?

Then you must investigate further.

How?

One way is to compare to the harmonic series, which diverges. Its terms have ratio or roughly . This ratio is not good enough for convergence. (and neither is for any c.)

What's wrong with the harmonic series?

Notice that this series has one term 1 that is at least 1, and one that is at least (namely ), two others that are at least ( and ), four others that are at least ( through ), others that are at least ( through ), and so on. Each of these groups sums to at least , so we get an endless number of 's which must diverge.

Suppose, for simplicity, that the terms in your series are arranged in decreasing order. (If not rearrange them.) You can make a histogram, creating blocks, each of width 1 and height between j - 1 and j given by the j-th term in your series. Then the value of the series is the area between the x axis and the top line of this histogram, between 0 and infinity.

You can then, with reasonable luck, be able to define a nice smooth curve that meet the left topmost points of each of your blocks, and another (the same one moved a distance one to the right) that meets the right topmost points of each of your blocks.

In the case of the harmonic series, these are the curves and .

When you do this you will notice that the first of these curves lies entirely under the top of your histogram, while the second lies entirely above it, except for its first term.

This means that the area under your histogram above the x axis is sandwiched between the area under your two curves, above the x axis (all for positive x, starting the upper curve at 1 and adding the first term to it).

The area under a curve is usual called the integral of the function defined by the curve, and we will study it soon and find ways to evaluate such areas. The series will converge when the areas under such curves are finite, and diverge otherwise, and that is a powerful criterion for absolute convergence. It is called the "integral test".

In the case of the harmonic series, the area under betweeon x = 0 and x = y can be shown to be ln(y+1) which is unbounded when y increases.