James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

742 Chapter 11 Infinite Sequences and Series
Rearrangements
The question of whether a given convergent series is absolutely convergent or conditionally
convergent has a bearing on the question of whether infinite sums behave like
finite sums.
If we rearrange the order of the terms in a finite sum, then of course the value of the
sum remains unchanged. But this is not always the case for an infinite series. By a rearrangement
of an infinite series o a n we mean a series obtained by simply changing the
order of the terms. For instance, a rearrangement of o a n could start as follows:
It turns out that
a 1 1 a 2 1 a 5 1 a 3 1 a 4 1 a 15 1 a 6 1 a 7 1 a 20 1 ∙ ∙ ∙
if o a n is an absolutely convergent series with sum s,
then any rearrangement of o a n has the same sum s.
However, any conditionally convergent series can be rearranged to give a different sum.
To illustrate this fact let’s consider the alternating harmonic series
6
1 2 1 2 1 1 3 2 1 4 1 1 5 2 1 6 1 1 7 2 1 8 1 ∙ ∙ ∙ − ln 2
(See Exercise 11.5.36.) If we multiply this series by 1 2 , we get
1
2 2 1 4 1 1 6 2 1 8 1 ∙ ∙ ∙ − 1 2 ln 2
Adding these zeros does not affect
the sum of the series; each term in the
sequence of partial sums is repeated,
but the limit is the same.
Inserting zeros between the terms of this series, we have
7
0 1 1 2 1 0 2 1 4 1 0 1 1 6 1 0 2 1 8 1 ∙ ∙ ∙ − 1 2 ln 2
Now we add the series in Equations 6 and 7 using Theorem 11.2.8:
8
1 1 1 3 2 1 2 1 1 5 1 1 7 2 1 4 1 ∙ ∙ ∙ − 3 2 ln 2
Notice that the series in (8) contains the same terms as in (6) but rearranged so that one
neg ative term occurs after each pair of positive terms. The sums of these series, however,
are different. In fact, Riemann proved that
if o a n is a conditionally convergent series and r is any real number whatsoever,
then there is a rearrangement of o a n that has a sum equal to r.
A proof of this fact is outlined in Exercise 52.
1. What can you say about the series o a n in each of the following
cases?
a
(a) lim
nl`
Z
n11
a
a n
Z − 8 (b) lim
n
Z
n11
l ` a n
Z − 0.8
2–6 Determine whether the series is absolutely convergent or
conditionally convergent.
s21d
2. ò
n21
n−1 sn
(c) lim
n
Z an11
l ` a n
Z
− 1
s21d
3. ò
n
n−0 5n 1 1
s21d
4. ò
n
n−1 n 3 1 1
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