Advanced High-School Mathematics

278 CHAPTER 5 Series and Differential Equations

We consider a couple of simple illustrations of the above theorem.

Example 1.The series

∑∞

n=1

(−1)n

n^2

will converge by the above theorem,

together with thep-Test.

Example 2. The series

∑∞

n=0

(−1)n

n

does not converge absolutely; how-

ever, we’ll see below that this series does converge.

An infinite series

∑∞

n=0

an which converges but is not absolutely con-

vergent is calledconditionally convergent. There are plenty of con-
ditionally convergent series, as guaranteed by the following theorem.

Theorem. (Alternating Series Test) Let a 0 ≥a 1 ≥a 2 ≥ ··· ≥ 0

and satisfy nlim→∞an = 0. Then the “alternating series”

∑∞

n=0

(−1)nan

converges.^13

We’ll conclude this section with two illustrations of theAlternating
Series Test.

Example 3. We know that the harmonic series

∑∞

n=1

1

n

diverges; how-

ever, since the terms of this series decrease and tend to zero, theAl-

ternating Series Testguarantees that

∑∞

n=1

(−1)n−^1

n

converges. We’ll

show later on that this actually converges to ln 2 (see page 302).

Example 4. The series

∑∞

n=0

n

n^2 + 1

can be shown to diverge by applying

theLimit Comparison Testwith a comparison with the harmonic

(^13) The proof of this is pretty simple. First of all, note that the “even” partial sums satisfy
(a 0 −a 1 )≤(a 0 −a 1 ) + (a 2 −a 3 )≤(a 0 −a 1 ) + (a 2 −a 3 ) + (a 4 −a 5 )≤···,
so it suffices to show that these are all bounded by some number (see Figure 1, page 266). However,
note that
a 0 −((︸a 1 −a 2 ) + (a 3 −a 4 ) + (a 5 −︷︷a 6 ) +···+ (a 2 n− 3 −a 2 n− (^2) ︸)
This is positive!
)−a 2 n− 1 ≤a 0 ,
so we’re done.