1549901369-Elements_of_Real_Analysis__Denlinger_

8.3 Series with Positive and Negative Terms 477

ALTERNATING SERIES

Definition 8.3.1 If {an} is a sequence of positive numbers, then both of the

00 00

series 2:)-l)n+lan and l.,)-l)nan are called alternating series.

n=l n=l

Since it is more common to write alternating series starting with a positive
term, we shall usually use the first of these two forms to represent a generic
alternating series. It will be clear in what follows that our results can be made
to apply to alternating series of the second form as well. There is a simple test
for determining whether certain alternating series converge.

Theorem 8.3.2 (Alternating Series Test) If {an} is a monotone decreas-
oo
ing sequence of positive numbers, then the alternating series l.,)-1)n+^1 an con-
n=l
verges if and only if an --> 0.

Proof. Let {an} be a monotone decreasing sequence of positive numbers.
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Part 1: Suppose l._)-1 r+l an converges. Then, by Theorem 8.1.4 (the
n=l
general term test), (-1r+^1 an--> 0, so an--> o.

Part 2: Suppose an--> 0. Consider the even-numbered partial sums, B2n =
2n
2:)-l)k+^1 ak. Observe that {S2n} is monotone increasing, since
k=l

S2(n+1) - S2n = [S2n + a2n+1 - a2n+2] - S2n

?:: 0, because {an} is monotone decreasing.

Next, observe that {S 2 n} is bounded above, since

S2n = a1 - a2 + a3 - a4 + · · · + a2n-l + a2n

= a1 - (a2 - a3) - (a4 - a5) - · · · - (a2n-2 - a2n-d - a2n

::; a 1 , since {an} is monotone decreasing and positive.

Therefore, { S 2 n} converges, by the monotone convergence theorem (2.5.3).

Now, S2n+I = S2n - a2n+I· Thus, by the algebra of limits of sequences,
{ S2n+i} converges, and

n-+oo lim S2n+1 = n-+oo lim S2n - n-+oo lim a2n+1 = n-+oo lim S2n·