18.2 Innite Geometric Series 571
P R O B L E M S F O R S E C T I O N 1 8 . 2
For Problems 1 through 11, determine whether the series converges or diverges. If it converges, nd its sum.
1. 1 10 100 10
2. 0.3 0.03 0.003 0.0003 0.00003
3. 3 9 27 3
4. 2 2 6 23
5. 1 2 4 8 16
6. 4 8 16 32
7. 3 1 2 4
8. 2 4 8 113
9. 1 1 2 3
10. 2 22 23 2
11. 22 23 24 2
12. Find the sum of the following. (If there is no nite sum, say so.)
(a) 3 9 27 320
(b) 3 3 3 3
(c) 0.2100.21000.21000 (d) 3 30.830.82 30.83
(e) 0.20.21.30.21.32 0.21.33 (f) 1 2 4 6 for 1 1
13. Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges.
(a) 3 2 16 128
1 1.1 1.21 1.331 100 1002 1003 1004
(c) 10000 11000 12100 13310 (d) 1 2 3 for 1
14. Write each of the following series rst as a repeating decimal and then as a fraction.
(a) 2 10 100 1000 (b) 3 102 104 12
572 CHAPTER 18 Geometric Sums, Geometric Series
18.3 A MORE GENERAL DISCUSSION OF INFINITE SERIES
In the previous four sections we focused on geometric sums and geometric series. In this section we broaden our discussion to investigate other innite series. Our focus in this chapter is geometric series, but you will have a better appreciation of geometric series if you have some familiarity with series that are not geometric.
Given an innite series
1 2 3
the most basic question to consider is whether the series converges or diverges.
Suppose all the terms of the innite series are positive. Then , the sum of the rst terms, is an increasing function. We know from our study of functions that an increasing function may increase without bound, or it may increase but be bounded, in which case it will be asymptotic to a horizontal line. In the latter case, the function must be increasing at a decreasing rate; in fact, if the function has a horizontal asymptote, its rate of increase must be approaching zero. Similarly, if lim for some nite constant , then the rate at which is increasing must be approaching zero. This translates to the observation that if an innite series is to have any chance at converging, then its terms must
be approaching zero, that is, lim
partial sums
must be 0.
partial sums
n
The partial sums are increasing at a decreasing rate;
The partial sums are bounded.
n
The partial sums are increasing at a decreasing rate;
The partial sums are unbounded.
Figure 18.1
If all of the terms of the series are negative, an analogous argument can be made. If some of the terms of an innite series are positive and others are negative, it is still true that in order for the series to have any shot at converging the terms must be approaching zero. If
lim , where 0, then the partial sums will be eventually increasing without bound if 0 and eventually decreasing without bound if 0. 7
Suppose the terms of a series are approaching zero; is this enough to guarantee convergence? In the case of geometric series the answer is yes, but in the general case the answer is NO! The situation in general is much more subtle;8 the next example will convince you of that fact.
7 If lim does not exist, then the partial sums will be bouncing around and will not converge.
8 We know that a function can be increasing at a decreasing rate and have a horizontal asymptote, but it can also be increasing
at a rate tending toward zero and yet be unbounded. Consider, for example, ln . Its rate of increase is given by . lim 0, so the rate of increase of ln tends toward zero as increases without bound. Nevertheless, as we know that ln .
18.3 A More General Discussion of Innite Series 573
EXAMPLE 18.7
SOLUTION
Does the innite series 2 3 4 5 converge or diverge? This innite series is called the harmonic series.
The terms of this series are going toward zero: lim 1 0. The harmonic series is not a geometric series. Can we put into closed form? Lets look at some partial sums.
1 2
1 1 5 2 3 6
1 1 1 13 2 3 4 12
1 1 1 1 77 2 3 4 5 60
2 3 4 ??
Closed form allowed us to get a rm grip on something rather slippery. In this example our luck has run out. We cannot express the general partial sum in closed form. The expression lim 1 1 1 1 gives us no insight into the convergence or divergence of the series. We need to take a different perspective. We will compare this series to a familiar series.
1 1
2 2
1 1 1 4 4 2
8 8 8 8 2
8 16 2
16 32 2
so
so
so
so
1 1 1 3 4 2
1 1 1 1 1 5 6 7 8 2
9 10 11 16 2
17 18 19 32 2
and so on . Think about this in terms of slices of pies. How many pies must we bake in order to give out slices as dictated by the harmonic series? If the series converges we need only bake some nite number of pies.
2 3 4 5 6 7 8 9 16 equals
2 3 4 5 6 7 8 9 16 half a more than more than more than
pie half a pie half a pie half a pie
which is greater than or equal to 2 2 2 2 . But this latter series diverges; consequently, the same must be true of the harmonic series. By comparing the harmonic
574 CHAPTER 18 Geometric Sums, Geometric Series
series with the divergent series 1 1 1 1 , we see that the harmonic series must
diverge. (No matter how many pies we bake, we will eventually run out and need more.)
We had previously observed that if an innite series is to have any chance at converg-ing, then the terms must be going toward zero. Although this is a necessary condition for convergence, a look at the harmonic series shows that it is not enough to guarantee conver-gence; the condition lim 0 is necessary for convergence but not sufcient.9 It is important to realize that if the terms of an innite series are going to zero, then the series may converge (as is true for all geometric series), yet on the other hand, the series may diverge (as in the example of the harmonic series).
A Summary of the Main Principles The Nth Term Test for Divergence:
If lim 0, then the series 1 2 3 diverges.10
Warning: This is a test for divergence only!
IncreasingandBoundedPartialSumsTest:Supposethetermsofaseriesareallpositive. Then increaseswith.Ifthepartialsumsarebounded, thatis, thereexistsaconstant such that for all , then it can be shown that lim exists and is nite. Therefore, the series converges.
In many cases the question of convergence or divergence of an innite series is a very subtle one. Often, for instance, one can determine that a certain series converges without being able to say exactly what it converges to. We will return to innite series in Chapter 30. Questions of convergence become much simpler if we focus on the special case of the geometric series, and this is our main focus in this chapter. In the case of geometric series,
convergence and divergence are straightforward to establish.
2 converges to 1
for 1 for 1.
In Section 18.4 we introduce some convenient notation for working with series, and in Section 18.5 we apply geometric series to real-world situations.
P R O B L E M S F O R S E C T I O N 1 8 . 3
ForProblems1through9, determinewhethertheseriesconvergesordiverges.Explain your reasoning.
1. 1 2 3 11
2. 1000 1000 1000 1000
3. 3 4 5
9 Necessary versus sufcient: In order for a polygon to be called a square it is necessary that it have four sides, but this alone is not sufcient to classify it as a square. In order to win a race it is necessary to nish it, but this alone is not sufcient.
10 For an innite series to diverge it is sufcient that lim 0. This, however, is not necessary. To lose a race it is sufcient not to nish it. That, however, is not the only way to lose a race!
18.4 Summation Notation 575
4. 222 323 424 515
(Hint:Comparethisterm-by-termtoageometricseriesyouknow.Chooseaconvergent
geometric series whose terms are larger than the terms of this series.)
5. sin 12 sin 22 sin 32 sin 2
(Hint:Comparethisterm-by-termtoageometricseriesyouknow.Chooseaconvergent geometric series whose terms are larger than the terms of this series.)
6. 3 4 5
7. 2 3 4 5
8. 2 2 2 2
9. 1 11 1 2 1 3 1
10. The sum 1 2 3 4 5 1 is not geometric, but we can express it in an easy-to-compute form.
Let
1 2 3 4 5 . (18.1)
Writing the terms from largest to smallest gives
122 1. (18.2)
Add equations (18.1) and (18.2) and divide by two to show that
1 2 3 4 5 1.
11. Challenge: Use the same line of reasoning as outlined in Problem 10 to show that
1 3 5 7 212.
12. Give an example of each of the following.
(a) An innite series that converges and whose partial sums are always increasing
(b) An innite series that converges and whose partial sums oscillate around the sum of the series
(c) An innite series that diverges although its terms approach zero
(d) An innite series that diverges but whose partial sums do not grow without bound
18.4 SUMMATION NOTATION
Sumswhosetermsfollowaconsistentpatterncanoftenbewritteninamorecompactwayby using summation notation. Summation notation is only notation; it is a compact shorthand for writing out a sum. We will introduce it via examples.
1 2 3 28
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