Main

Absolute And Conditional

Up until now, the convergence tests covered by this course have only covered series with positive terms. What happens when a series has some positive and some negative terms? This module describes some tools for determining the convergence or divergence of such a series.

Alternating series test

One particular type of series is fairly simple to test for convergence. A series is alternating if it is of the form , where is a positive sequence. In other words, a series is alternating if its terms are alternately positive and negative.

Alternating Series Test

An alternating series with decreasing terms converges if and only if .

The intuition behind this test is that if the terms are alternating, decreasing, and go to zero, then the partial sums of the series gradually zero in on the true value. The first partial sum is greater than the true value, the second partial sum is less than the true value, and so on:

Example

The series

is alternating. Its terms are decreasing, and the terms go to zero, so by the alternating series test, the series converges.

Example

Determine if

converges.

The series is alternating, the terms are decreasing, and . Thus, by the alternating series test, the series converges.

Alternatively, we can observe that this series is geometric:

Example

Determine if

converges.

The series is alternating, terms are decreasing, and

And so by the alternating series test, the series converges.

Conditional and absolute convergence

Consider the two previous examples. Note that both and converge. However, if these series were not alternating, then still converges (it is geometric), whereas does not converge (comparison with the harmonic series ).

These series demonstrate a distinction between two types of convergence.

Absolute and Conditional Convergence

The series converges absolutely if converges.

A series converges conditionally if converges, but diverges.

In other words, a series is conditionally convergent if it is convergent but not absolutely convergent.

Example

Using this terminology, the series

converges absolutely, but

converges conditionally.

Example

Find the values of for which the alternating p-series

diverges, converges conditionally, and converges absolutely.

When , the terms of the series do not go to 0. That is,

So by the nth term test for divergence, the series diverges for these values of .

When , the terms of the series are decreasing, alternating, and going to 0. So by the alternating series test, the series converges for this range of . However, if absolute values are taken, then the resulting series, , does not converge by the p-series test. Thus, for , the series converges conditionally.

When , the series converges absolutely by the p-series test.

Example

Determine if

converges, and if so, whether it is conditional or absolute convergence.

The series is alternating, but notice that the terms do not go to 0, since . Therefore, by the nth term test for divergence, this series diverges.

Example

Determine if

converges, and if so, whether it is conditional or absolute convergence.

The series is alternating, and note that , which leads us to believe that the series converges. In fact, the series converges absolutely, i.e.

converges, as the ratio test shows, along with some careful algebra:

So , and so the series converges absolutely.

Absolute convergence test

Some series are not strictly alternating, but have some positive and some negative terms, sporadically. In this situation, it can be difficult to determine whether the series converges directly, but the following test sometimes makes the determination easier.

Absolute Convergence Test

If the series converges, then converges. In other words, if a series converges absolutely, then the series converges.

Consider the series . Note that

So , and by the comparison test we find converges. Then

converges too, being the difference of two convergent, positive series.

Example

Determine if

converges or diverges.

is a messy function because it is sometimes positive and sometimes negative, but not in a simple alternating pattern. However, one nice thing about is that, in absolute value, it is bounded by 1. So

Since converges by p-series test, it follows by the comparison test that converges. Hence converges absolutely, and so it converges by the absolute convergence test.


EXERCISES

  • Determine whether the following series converges or not. If it converges, is it conditionally convergent or absolutely convergent?

  • Determine whether the following series is convergent or divergent