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Discussion C7: What to Do in the Face of a Series
© 1994 by CaRP, Department of Mathematics, University of California, Davis

In today’s discussion, we will practice classifying an infinite series $sum oo a n=1 n$ into one of three categories:

Divergent: $oo sum an n=1$ diverges
Conditionally convergent: $oo sum n=1an$ converges, but $oo sum n=1|an|$ diverges.
Absolutely convergent: $oo sum |an| n=1$ converges. (Note: In this case, since $sum oo n=1|an|$ converges, so does $sum oo n=1$ $an$ , by the Absolute Convergence Test.)

An important distinction between an absolutely convergent series and a conditionally convergent series is the fact that the terms of an absolutely convergent series can be rearranged in any order without affecting the sum of the series, whereas the terms of a conditionally convergent series can be rearranged so that the resulting series converges to any specified number or even diverges.

The following is an outline of a basic strategy for determining whether a given series $oo sum a n=1 n$ converges absolutely, converges conditionally, or diverges.

BASIC STRATEGY

As a general rule of thumb, try the nth Term Divergence Test first.
- If $lni-->mo o an /= 0$ , then we can immediately conclude that $oo sum n=1 an$ diverges.
- On the other hand, if $nli-->mo o an = 0$ , then no conclusion can be drawn at this point.
If the th Term Divergence Test is inconclusive, the next step is to examine the corresponding series . At this point, one has a variety of tests that may or may not apply to .
- Integral Test: especially useful if $oo {|an |}n=1$ is eventually decreasing and the function that is obtained by substituting $x$ for each $n$ in $|an|$ is easy to integrate.
- Limit Comparison Test: especially useful if $|an |$ is a rational function in $n$ or in some root of $n$ .
- Comparison Test: especially useful if $|an |$ involves either a sine or a cosine.
- Ratio Test: especially useful if $|an|$ has factorials in products somewhere, or if it looks like the product of a rational function and an exponential.
- Root Test: especially useful if it’s easy to calculate the nth root of $|an|$ .
Depending on the results of these tests, we can make the following assertions:
- If $oo sum |an | n=1$ converges by any of the above tests, then by an application of the Absolute Convergence Test, we know that $sum oo an n=1$ converges as well; in this case, we conclude that $oo sum a n=1 n$ converges absolutely, and we are done.
- If $oo sum |an | n=1$ diverges by use of the Ratio Test or the Root Test, then $oo sum n=1an$ also diverges. (Think about it. Applying the Ratio or Root Test to $oo sum |an| n=1$ is the same as applying the Absolute Ratio or Absolute Root test to $oo sum a n=1 n$ .) We are done.
- But if $oo sum |an| n=1$ diverges by the Integral, Comparison, or Limit Comparison Test then the best possible conclusion is that $oo sum an n=1$ cannot converge absolutely; in this case, we can assert that either $oo sum an n=1$ converges conditionally or diverges. Since we don’t know which, we aren’t done yet.
If an analysis of $sum oo |an| n=1$ yields ambiguous results about $sum oo an n=1$ (i.e. that either $oo sum a n=1 n$ diverges or converges conditionally), then we must examine $oo sum a n=1 n$ itself. For example, if $oo sum an n=1$ is an alternating series, $nl-->imo o an = 0$ , and $|an|$ decreases, then the Alternating Series Convergence Test applies.

As the above strategy indicates, to be successful at classifying a given series, you must know your series tests. The problems in the self-help section are designed to refresh your memory with regards to the various tests.

Self-Help Background Check

In order to get the most benefit from the discussion period, do the following problems before the start of discussion. You can check your answers with those given after the end of the discussion assignment.

Consider the series . In each case, determine if the indicated test tells us whether the series converges or diverges or yields inconclusive results.
1. The nth term divergence test.
2. The Integral Test.
3. The Comparison Test. (Compare to the series $oo sum 1- n n=1$ .)
4. The Limit Comparison Test. (Compare to the series $oo sum 1- n=1 n$ .)
5. The Ratio Test.
Consider the series . Determine whether the indicated test tells us the series converges or diverges or yields inconclusive results.
1. The Alternating Series Test.
2. The Absolute Convergence Test.

If you missed more than one problem, consult our text on page 617, or if time permits, go to office hours for clarification. In addition, problems 6 and 8 on page 618 (GUIDE QUIZ) provide extra practice.

Guided Inquiry

Consider the series $sum oo n n2-+-1- n=1(- 1) n3 + 2$ .

Fill in what’s missing to determine whether it diverges, converges absolutely, or converges conditionally.

The first test we should try is the $----------------------------$ Test. We see that $nli-->mo o an =-------$ since the degree of the numerator is less than the degree of the denominator. Thus, the $n$ th Term Divergence Test is inconclusive.

Since the nth Term Divergence Test is inconclusive, we should examine the corresponding series $sum oo |an| n=1$ . In this case, we have that

sum oo oo sum |an|= . n=1 n=1--------------

Since this series is a rational function in $n$ , the most useful test is the
$---------------------$ Test. Comparing $oo 2 sum n--+-1- n=1n3 + 2$ to the series $oo sum 1- n=1 n$ , we note that

lim _n	= lim _n
	= _________.

Since this limit is finite and nonzero and since $oo sum 1- n=1 n$ diverges (by the
$--------------$ Test), we conclude that the series $2 oo sum |a |= oo sum n--+-1- n=1 n n=1 n3 + 2$ must also $--------------$ , by the $----------------------------$ Test. Therefore,
$oo sum oo sum n2 + 1 an = (- 1)n------- n=1 n=1 n3 + 2$ cannot converge absolutely; it must either diverge or converge $--------------$ .

Finally, we apply the $----------------------------$ Test to the series $2 oo sum a = oo sum (- 1)n n-+-1- n=1 n n=1 n3 + 2$ . First, however, we must verify that this series satisfies the hypotheses of the alternating series test. To refresh our memory, let’s restate the hypotheses of the test:

$lim an= ------- n-->o o$
The sequence ${|an| }$ is $--------------$ , for $n$ sufficiently large.
$sum oo n=1an$ is an alternating series.

Now, examine each of these for the series under consideration.

We already checked in part (a) that $n2-+-1- nl-->imo o n3 + 2 = 0$ . Moreover, the sequence ${ n n2 + 1} (- 1) -3----- n + 2$ clearly alternates because $(-1)n$ alternates in sign and $2 n--+-1- n3 + 2$ is always positive (since $n > 1$ ). Thus, it suffices to check that the sequence ${ } n2-+-1- n3 + 2$ is decreasing.

One way to do this is to show that the function $2 f(x) = x--+-1- x3 + 2$ is decreasing for $x$ sufficiently large, or equivalently that $f '(x) < -------$ , for $x$ sufficiently large. But

f^'(x)	=
	=
	= .

It is easy to see that for $x > 2$ , we have $f '(x) < 0$ . Thus, $f (x)$ decreases. So the sequence ${ n2 + 1} ------- n3 + 2$ must also decrease. Since $oo sum sum oo n2 + 1 an = (- 1)n ------- n=1 n=1 n3 + 2$ satisfies all the conditions to the $----------------------------$ Test, we assert that this series converges.

In part (b), we showed that $oo sum oo sum 2 |an|= n--+-1- n=1 n=1 n3 + 2$ diverges; but in part (c), we showed $oo sum sum oo n n2-+-1- an = (- 1) n3 + 2 n=1 n=1$ converges. So, by definition, we conclude that $sum oo oo sum n n2 + 1 an = (- 1) -3----- n=1 n=1 n + 2$
converges $----------------------------$ .

Group Investigation

If you want to be successful in dealing with problems involving absolute and conditional convergence, you must learn to recognize which convergence tests apply in which situations. The following problem is designed to give you some practice in quickly identifying the most effective convergence/divergence test.

Using the guidelines outlined in the basic strategy, discuss with the other members of your group which test you would use for each of the following series and whether you think the series converges or diverges.
1. $3 oo sum ----n-+--2--- n=1n5 - 4n2 + 1$
2. $oo sum -n2- n=110n$
3. $oo sum -n!- n=110n$
4. $oo sum --sin2-n-- n=1n(n + 4)$
5. $oo sum ---1--- n=1n1 -1/n$
6. $oo sum ---1---- n=1n (ln n)2$

Discussion C7 Assignment

At the start of class on the day indicated by your instructor, hand in your well-written work for the following problems, Refer to the “Mathematical Writing” section in Chapter 1 of Cameos for guidance on style. Feel free to discuss the problems with others, BUT your solutions should be written independently in your own words.

Determine if the given series diverges, converges absolutely, or converges conditionally.
1. $oo sum 1 (- 1)n- V~ ---- n=1 n ln n$
2. $oo 3n sum (- 1)n--n---- n=1 4n(n!)3$
Let be a series whose terms may be either positive or negative.
1. Suppose $oo sum an n=1$ converges absolutely. Is it necessarily true that $oo sum n=1$ $a2n$ converges absolutely? If so, provide a proof of this claim. Otherwise, provide a counterexample.
2. Suppose $oo sum an n=1$ converges conditionally. Is it necessarily true that $sum oo a2n n=1$ converges conditionally? If so, provide a proof of this claim. Otherwise, provide a counterexample.

Brief answers to problems from the Self-Help Background Check

1. No conclusion can be made since $n lni-->m oo n2-+-1-= 0$
2. Diverges.
3. Since $n 1 --2----- < -- (n + 1) n$ and $oo sum 1 -- n=1 n$ diverges, no conclusion can be made.
4. Since $2 lim n/-(n-+--1)= 1 n-->o o 1/n$ and $oo sum 1- n=1 n$ diverges, we conclude that the given series also diverges.
5. Ratio Test yields no conclusion.
1. Converges.
2. No conclusion can be made.

Mathstory

The Bernoulli brothers, Jakob (1654-1705) and Johann (1667-1748), played central roles in the development of Calculus. Both were self-taught mathematicians who corresponded with Leibniz, became his students, and subsequently championed Leibniz’ view of Calculus. Jakob rose to appointment as a professor of mathematics in the delightful and beautiful Swiss city of Basel; Johann became a professor of mathematics at Groningen in Holland. After Jakob’s death, Johann succeeded him in the chair at Basel. Jakob’s work included the invention of polar coordinates, the study of curves such as the lemniscate (given by $r = 1/ cosh$ ) and exponential spiral, and infinite series. He established the divergence of the harmonic series and the convergence of the $p$ series, where $p = 2$ . The actual sum of the latter series was a source of much discussion; it was not until 1736 that Johann’s student, Euler, discovered that it converges to $p2/6$ .

Literature Cited

Simmons, G. F. 1992. Calculus Gems: Brief Lives and Memorable Mathematics. McGraw-Hill, Inc. New York.

Discussion C7: What to Do in the Face of a Series© 1994 by CaRP, Department of Mathematics, University of California, Davis