Sequences and Series 1894.8 Some Properties of Series of Complex Terms
Theorem 4.8 The sum of an absolutely convergent series does not change
if the order of its terms is rearranged arbitrarily (commutative property of
absolutely convergent series).
A rearrangement of a series I: an is a series I: a~, where each an occurs
just once among the a~, and conversely.
Proof If I:~ an = L:~(bn + icn) converges absolutely, then both I:~ bn
and I:~ en converge absolutely (as noted above), and it is permissible to al-
ter the order of the terms without changing their sum (by the corresponding
theorem for real series). Hence the same property holds for I:~ an.Remarks I. If I:~ an is convergent, but not absolutely, the series I:~ bn
and :L~ Cn are convergent (Theorem 4.7), but both I:~ lbnl and I:~ lcnl
cannot converge, i.e., at least one is divergent, in view of the inequality
lanl :::; lbnl + lcnl· It follows that in this case, changing the order of the
terms of I:~ an may vary its sum, and may even produce divergence, in
particular, an oscillatory series.
IL Because the order of the terms of an absolutely convergent series can
be altered arbitrarily, such series are also called unconditionally conver-
gent. Those that converge, but not absolutely, are then called conditionally
convergent (or, simply convergent).Theorem 4.9 In a convergent series any finite number of consecutive
terms can be replaced by its sum without affecting the convergence of
the series or its sum; similarly for series that diverge to oo (associative
property for series).
Proof Consider the series I:~ an and suppose that we associate its terms
as follows:(4.8-1)
Let An denote the partial sums of I:~ an, and AAr the partial sums of
the series (4.8-1). SinceA~ =Ai> A~ = Ak, A; =Am,
it is clear that the sequence {AAr} is a subsequence of {An}, so it has the
same limit as {An}·
Note If I: an is oscillatory, the associative property does not hold.Example The series i-i+i-i+· ··is oscillatory, yet (i-i)+(i-i)+· · · =
0 + 0 + · · · converges.