20. Convergence of Positive Series

a. Overview

Given a series \(\displaystyle \sum_{n=n_o}^\infty a_n\), our goal is to find the sum of the series, either exactly (as for the geometric or telescoping series) or approximately (as done at the end of this chapter). To approximate the series, we usually just add up a finite number of terms. Once we do that, we will want to know the error in using a finite number of terms instead of an infinite number of terms. However, before we can even approximate the series, we at least need to know the series has a finite sum, i.e. that it converges. Otherwise, adding up a finite number of terms is a waste of time.

Checking a series for convergence is much like doing an integral. In doing an integral, you must look at the integrand and determine which integration technique to use. In checking a series for convergence, you must look at the terms and decide which convergence test to use. We split the convergence tests into two groups, those for positive series and those for more general series.

A series \(\displaystyle \sum_{n=n_o}^\infty a_n\) is positive if all of its terms are positive, \(a_n>0\) for all \(n\).
A series \(\displaystyle \sum_{n=n_o}^\infty a_n\) is negative if all of its terms are negative, \(a_n< 0\) for all \(n\).
A general series is an arbitrary series which is not necessarily positive and not necessarily negative.

Here, we list the Convergence and Divergence Tests for Positive Series along with the necessary definitions. There is no attempt to justify or prove the tests. That will be done later in this chapter. If a series is negative, the tests for positive series will work after an overall minus sign is factored out. At the end of the section, we will come back to approximating the series by a finite number of terms and finding the error in this approximation.

We start with the two tests covered in the previous chapter.

A tail of a series \(\displaystyle \sum_{n=n_o}^\infty a_n\) is any series of the form \(\displaystyle \sum_{n=N}^\infty a_n\) where \(N>n_o\).

A series \(\displaystyle \sum_{n=n_o}^\infty a_n\) is convergent if and only if any (and hence every) tail is convergent.

This proposition says the convergence of a series does not depend on any finite number of terms. Further, if the tail is all positive (or all negative) you can apply the tests for positive series to the tail.

Don't forget to apply the \(n^\text{th}\) Term Divergence Test. Then move on to the other tests.

If \(\displaystyle \lim_{n\to\infty}a_n\neq 0\), then \(\displaystyle \sum_{n=n_o}^\infty a_n\) is divergent.

If \(\displaystyle \lim_{n\to\infty}a_n=0\) the \(n^\text{th}\) Term Divergence Test FAILS and says nothing about \(\displaystyle \sum_{n=n_o}^\infty a_n\).

If \(a_n=f(n)\) where \(f(x)\) is a continuous, positive, decreasing function on \([n_o,\infty)\), then \(\displaystyle \sum_{n=n_o}^\infty a_n\) is convergent if and only if \(\displaystyle \int_{n_o}^\infty f(x)\,dx\) is convergent.

In practice, we don't switch from the terms \(a_n\) to the function \(f(x)\). We just regard \(n\) as a continuous variable and \(a_n\) as a continuous function of \(n\) and compute \(\displaystyle \int_{n_o}^\infty a_n\,dn\)

As a special case of the integral test we have:

The \(p\)-series \(\displaystyle \sum_{n=n_o}^\infty \dfrac{1}{n^p}\) is convergent if \(p \gt 1\) and is divergent if \(p \le 1\).

In applying the simple comparison test, \(\displaystyle \sum_{n=n_o}^\infty a_n\) is the original series while \(\displaystyle \sum_{n=n_o}^\infty b_n\) is the comparison series whose convergence is already known.

Sometimes we want to compare an original series \(\displaystyle \sum_{n=n_o}^\infty a_n\) to a comparison series \(\displaystyle \sum_{n=n_o}^\infty b_n\) but we can't prove the necessary inequality. In that case, we should try the limit comparison test:

Cases (2) and (3) are called the extreme cases, and arise very rarely.
If \(L=0\) and \(\displaystyle \sum_{n=n_o}^\infty b_n\) is divergent, or \(L=\infty\) and \(\displaystyle \sum_{n=n_o}^\infty b_n\) is convergent, the Limit Comparison Test FAILS.

If \(\rho=1\) the Ratio Test FAILS and says nothing about \(\displaystyle \sum_{n=n_o}^\infty a_n\).

If \(\rho=1\) the Root Test FAILS and says nothing about \(\displaystyle \sum_{n=n_o}^\infty a_n\).