19 Convergence
Highlights of this Chapter: Finding the value of a series explicitly is difficult, so we develop some theory to determine convergence without explicitly finding the limit. Our main tool is comparison, which is built using the Monotone convergence theorem; and in particular comparison with a geometric series - the Ratio Test. Along the way to developing this theory we study a few important special series:
- We prove the harmonic series
diverges. - In contrast, we prove that the sum of reciprocal squares
converges. In the final project we will show its value is .
In this section, we build up some technology to prove the convergence (and divergence) of series, without explicitly being able to compute the limit of partial sums. Such results will prove incredibly useful, as in the future we will encounter many theorems of the form if
19.1 The Cauchy Criterion
For sequences, after some work we were able to find a definition equivalent to the original notion of convergence, which did not mention the precise value of the limit. This is exactly the sort of thing we seek for our investigation into series, so we carry it over directly here:
Definition 19.1 (Cauchy Criterion) A series
Exercise 19.1 Prove a series satisfies the Cauchy criterion if and only if its sequence of partial sums is a Cauchy sequence.
Because we know that being convergent and cauchy are equivalent, this means that all series that satisfy the Cauchy criterion are convergent, and conversely if a series does not, then it must diverge. We use this second observation to construct an easy-to-apply test for divergence:
Corollary 19.1 (Divergence Test) If a series
Proof. Let’s apply the cauchy condition to the single value
But making
This is useful mostly to immediately rule out the possibility that certain series converge. For instance it tells us that
19.1.1 Absolute Convergence
Below we will develop several theorems that apply exclusively to series of positive terms. That may seem at first to be a significant obstacle, as many series involve both addition and subtraction! So, we take some time here to assuage such worries, and provide a means of probing a general series using information about its nonnegative counterpart.
Definition 19.2 (Absolute Convergence) A series
Of course, such a definition is only useful if facts about the nonnegative series imply facts about the original. Happily, that is the case.
Theorem 19.1 (Absolute Convergence Implies Convergence) Every absolutely convergent series is a convergent series.
Proof. Let
But, by the triangle inequality we know that
19.2 Comparison
One of the very most useful convergence tests for a series is comparison. This lets us show that a series we care about (that may be hard to compute with) converges or diverges by comparing it to a simpler series - much like the squeeze theorem did for us with sequences. This theorem gives less information than the squeeze theorem (it doesn’t give us the exact value of the series we are interested in) but it is also easier to use (it only requires a bound, not an upper and lower bound with the same limit).
Theorem 19.2 (Comparison For Series) Let
- If
converges, then converges. - If
diverges, then diverges.
The proof is just a rehashing of our old friend, Monotone Convergence.
Proof. We prove the first of the two claims, and leave the second as an exercise. If
Thus.
Exercise 19.2 Let
The comparison test is incredibly useful: two of the most famous series it lets us understand are left as exercises below.
Exercise 19.3 Prove that
Exercise 19.4 Show the harmonic series
19.3 The Ratio Test
We saw in the last chapter that geometric series - where the consecutive ratios of every pair of terms is constant - are particularly easy to sum. Now that we have comparison, we can leverage this to provide a powerful convergence test for a much larger collection of series: those whose consecutive rations are constant in the limit.
Theorem 19.3 (The Ratio Test) Let
Proof. We prove the convergence claim for
Assume that
But the first finitely many terms of a series cannot affect whether or not it converges, so we see that
This is the definition of
Exercise 19.5 Prove that if
Note that this test does not tell us anything when
Example 19.1 The sequence
But, the sequence
Remark 19.1. There is an even more general version of the ratio test were we don’t assume that
Exercise 19.6 Prove that the following series converges:
19.4 Other Convergence Tests
Because series are ubiquitous throughout mathematics, there are many more convergence theorems that have been developed than we have the time to cover here. Though we will not need them in our course, I list two of the most popular (following the ratio test) below for reference.
Theorem 19.4 (The Root Test) Let
The following test shows up in a Calculus II course; though we are not ready to rigorously discuss it yet as it requires integration. Once we gain some ability with integrals, this will allow us to leverage our abilities with the Fundamental Theorem to prove new facts about series.
Theorem 19.5 (The Integral Test) If
19.5 Conditionally Convergent Series
Definition 19.3 A series converges conditionally if it converges, but is not absolutely convergent.
Such series caused much trouble in the foundations of analysis, as they can exhibit rather strange behavior. We met one such series in the introduction, the alternating sum of
19.5.1 Alternating Series
Definition 19.4 (Alternating Series) An alternating series is a series of the form
Theorem 19.6 (Alternating Series Test) If
Before jumping in, its helpful to take a look at a few partial sums to start. For example,
Grouping the terms of this finite sum like so shows that
Exercise 19.7 Let
- All the partial sums
are nonnegative. - All partial sums are bounded above by the first term
.
Corollary 19.2 Starting the sum at
What other patterns can we notice? Increasing from
Exercise 19.8 Let
- The even subsequence is monotone decreasing
- The odd subsequence is monotone increasing
Because each of these subsequences is monotone and bounded (by the previous exercise) they converge via monotone convergence. Now, all we need to see is they converge to the same limit to assure convergence of the entire series, by Theorem 11.2.
Proposition 19.1 Let
Proof. Let
19.5.2 Properties of Conditionally Convergent Series
First we look at the main example of a conditionally convergent series.
Example 19.2
- It converges, by the alternating series test.
- But it is not absolutely convergent, as
diverges by EXR
This series is famous from the introduction to our course, where we saw that its value when summed is the natural logarithm of 2, but that this value changes when we reorder the terms! This is a general behavior of conditionally convergent series; and one hint of this is that the sum of their positive and negative terms separately each diverges to
Theorem 19.7 If
For an absolutely convergent series, this cannot happen, and the sums of all the positive terms converges, as does the sum of all the negative terms.
Exercise 19.9 Prove that if