Kenneth Kuttler

5.2. ABSOLUTE CONVERGENCE 85

Formula 5.3 follows from the observation that, from the triangle inequality,∣∣∣∣∣ n

∑k=m

∣∣∣∣∣≤ n

∑k=m|ak| ≤

∞

∑k=m|ak|

and so |∑∞k=m ak|= limn→∞ |∑n

k=m ak| ≤ ∑∞k=m |ak| .

Recall that if limn→∞ An = A, then limn→∞ |An|= |A|.

Example 5.1.6 Find ∑∞n=0( 5

2n +63n

From the above theorem and Theorem 5.1.4,∞

∑n=0

(52n +

63n

)= 5

∞

∑n=0

12n +6

∞

∑n=0

13n = 5

11− (1/2)

+61

1− (1/3)= 19.

The following criterion is useful in checking convergence. All it is saying is that theseries converges if and only if the sequence of partial sums is Cauchy. This is what thegiven criterion says. However, this is not new information.

Theorem 5.1.7 Let {ak} be a sequence of points in F. The sum ∑∞k=m ak converges

if and only if for all ε > 0, there exists nε such that if q≥ p≥ nε , then∣∣∣∑q

k=p ak

∣∣∣< ε.

Proof: ⇒Suppose first that the series converges. Then {∑nk=m ak}∞

n=m is a Cauchysequence by Theorem 4.5.3 on Page 62. Therefore, there exists nε > m such that if q ≥p−1≥ nε > m, ∣∣∣∣∣ q

∑k=m

ak−p−1

∑k=m

∣∣∣∣∣=∣∣∣∣∣ q

∑k=p

∣∣∣∣∣< ε. (5.4)

⇐Next suppose the other condition, that∣∣∣∑q

k=p ak

∣∣∣< ε whenever p,q sufficiently large.

Then from 5.4 it follows that {∑nk=m ak}∞

n=m is a Cauchy sequence and so, by Theorem4.8.14, it converges. By the definition of infinite series, this shows the infinite sum con-verges as claimed.

5.2 Absolute ConvergenceDefinition 5.2.1 The statement that a series ∑

∞k=m ak converges absolutely means

∑∞k=m |ak| converges. If the series does converge but does not converge absolutely, then it is

said to converge conditionally.

Theorem 5.2.2 If ∑∞k=m ak converges absolutely, then it converges.

Proof: Let ε > 0 be given. Then by assumption and Theorem 5.1.7, there exists nε

such that whenever q ≥ p ≥ nε , ∑qk=p |ak| < ε. Therefore, from the triangle inequality,

ε > ∑qk=p |ak| ≥

∣∣∣∑qk=p ak

∣∣∣ . By Theorem 5.1.7, ∑∞k=m ak converges.

In fact, the above theorem is really another version of the completeness axiom. Thusits validity implies completeness. You might try to show this.

One of the interesting things about absolutely convergent series is that you can “addthem up” in any order and you will always get the same thing. This is the meaning of thefollowing theorem. Of course there is no problem when you are dealing with finite sumsthanks to the commutative law of addition. However, when you have infinite sums strangeand wonderful things can happen because these involve a limit.

5.2. ABSOLUTE CONVERGENCE 85Formula 5.3 follows from the observation that, from the triangle inequality,n n fone}Ya] < YE lael < Yo Jaak=m k=mk=mand so Diem a| = limp yoo =m ag| < vim lax| |Recall that if limy_,..A, =A, then limp. |An| = |A|.Example 5.1.6 Find Dy (4+).From the above theorem and Theorem 5.1.4,a 5 6 a | a | 1 1—4+—)=5Y —+6Y —=5 +6 = 19.Lats) Sha he map manThe following criterion is useful in checking convergence. All it is saying is that theseries converges if and only if the sequence of partial sums is Cauchy. This is what thegiven criterion says. However, this is not new information.Theorem 5.1.7 Ler {ag} be a sequence of points in F. The sum Yr _,, ax convergesif and only if for all € > 0, there exists ng such that if q > p > ne, then tp ax| <€é.Proof: =Suppose first that the series converges. Then {Y7_,,ax}/_,,, is a Cauchysequence by Theorem 4.5.3 on Page 62. Therefore, there exists ng > m such that if g >p-l=>ne>m,<€. (5.4)q pol¥ aT a=k=m k=mqYak=p< Next suppose the other condition, that a p ax| < € whenever p,q sufficiently large.Then from 5.4 it follows that {Y7_,,ax};_,, is a Cauchy sequence and so, by Theorem4.8.14, it converges. By the definition of infinite series, this shows the infinite sum con-verges as claimed. §f5.2 Absolute ConvergenceDefinition 5.2.1 he statement that a series Vem converges absolutely meansYen |ae| converges. If the series does converge but does not converge absolutely, then it issaid to converge conditionally.Theorem 5.2.2 Tf Vieem & converges absolutely, then it converges.Proof: Let € > 0 be given. Then by assumption and Theorem 5.1.7, there exists ngsuch that whenever g > p => ne, Lie |a,| < €. Therefore, from the triangle inequality,€> LiL, lax| = fp an| . By Theorem 5.1.7, Y7_,,, 4, converges. WlIn fact, the above theorem is really another version of the completeness axiom. Thusits validity implies completeness. You might try to show this.One of the interesting things about absolutely convergent series is that you can “addthem up” in any order and you will always get the same thing. This is the meaning of thefollowing theorem. Of course there is no problem when you are dealing with finite sumsthanks to the commutative law of addition. However, when you have infinite sums strangeand wonderful things can happen because these involve a limit.