484 Chapter 8 • Infinite Series of Real Numberssecond negative term. Continue by induction. In general, after the nth negative
term has been added, add the next consecutive positive terms until the sum is
greater than n+ 1. Since this process accounts for all the terms of I: an, it yields
a rearrangement of I: an. This rearrangement diverges because its sequence of
partial sums is unbounded above.(b) Let r E !Et We shall construct a rearrangement of I: an that converges
to r. The following argument uses the fact that the series of positive terms of
Lan diverges to +oo and the series of negative terms of Lan diverges to - oo.
It also uses the general term test (8.1.4), assuring us that an---+ 0.
Add consecutive positive terms, but (only) enough positive terms so that
the sum is greater than r. Then add consecutive negative terms, but (only)
enough so that the cumulative sum is less than r. Continue, adding positive
terms, but (only) enough positive terms so that the cumulative sum is greater
than r. Then continue, adding consecutive negative terms, but (only) enough so
that the cumulative sum is less than r. Continue this process by mathematical
induction. The resulting series will use each term of Lan exactly once, and
thus be a rearrangement I: a"k of the series I: an. As noted above, an ---+ 0, so
a"n ---+ 0. (See Exercise 2.2.23.)
Because of the way in which we have constructed the rearrangement, the
n
partial sum Sn = L a"k never deviates from r by more than the absolute
k=l
value of the last term added. That is, 'v'n EN,
Therefore, by the second squeeze principle, S"n ---+ r. •
So, what is conditional about a "conditionally" convergent series? One
way of putting it is to say that the terms of an absolutely convergent series
can be added in any order (i.e., unconditionally) without affecting the sum. In
contrast, the terms of a conditionally convergent series can be added only on
the "condition" that we add them in the right order. If we alter that order we
may alter the sum or even lose convergence.The next theorem shows another contrast between absolutely and condi-
tionally convergent series. A conditionally convergent series always has at least
two divergent subseries; namely, the series of its positive terms and the series of
its negative terms. So, a subseries of a conditionally convergent series may di-
verge. However, we shall now show that an absolutely convergent series cannot
have a divergent subseries.
Definition 8.3.15 A subseries of a series I: an is a series of the form I: ank,
where {nk} is a subsequence of {n}.