482 Chapter 8 • Infinite Series of Real Numbers
REARRANGEMENTS AND SUBSERIESWe shall now explore some deeper consequences of absolute convergence,
and thereby see more clearly the difference between absolute and conditional
convergence.
Definition 8.3.12 A rearrangement of a series Lan is a series of the form
L aan' where(]" is a permutation^4 of the set of natural numbers and (Jn= (J"(n).
Theorem 8.3.13 Every rearrangement of an absolutely convergent series con-
verges absolutely, and has the same sum.
Proof. (a) Let L an be a convergent nonnegative series, with rearrange-
ment Laa", and denote their partial sums by
n n
Sn= L ak and Sn= L aak·
k=l k=lNow, \In EN, (]" 1 ,(]" 2 ,-··,(Jn E {1,2,-· · ,m}, where rri = max{(J"k k =
1, 2, · · · , n}. Thus, since the terms are nonnegative,
00
Sn :S Sm :S L ak.
k=lThus, {Sn} is a monotone increasing sequence with an upper bound, so it
must have a limit. And, since limits preserve inequalities,
00 00
L aak :S L ak.
k=l k=lWe have just shown t hat every rearrangement of a convergent nonnegative
series converges and has a sum less than or equ al to that of the original series.
But Lan is also a rearrangement of L aan. Applying the result just proved,
Lan :S L aan. Therefore, equality holds, and we have proved the desired result
for nonnegative series.
(b) From Part (a) we can conclude that every rearrangement of an abso-
lutely co nvergent series converges absolutely. It remains to prove that a rear-
rangement of an absolutely convergent series must have the same sum as the
original series.
- A permutation of a set A is a 1-1 correspondence a: A-+ A. See Appendix B.2.