Kenneth Kuttler

246 CHAPTER 13. SOME FUNDAMENTALS∗

Definition 13.8.1 The symbol ∑∞k=1f k (x) means limn→∞ ∑

nk=1f k (x) provided this limit

exists. This is called pointwise convergence of the infinite sum. Thus the infinite sum meansthe limit of the sequence of partial sums. The infinite sum is said to converge uniformly ifthe sequence of paritial sums converges uniformly.

Note how this theorem includes the case of ∑∞k=1ak as a special case. Here the ak don’t

depend on x.The following theorem is very useful. It tells how to recognize that an infinite sum is

converging or converging uniformly. First is a little lemma which reviews standard calcu-lus.

Lemma 13.8.2 Suppose Mk ≥ 0 and ∑∞k=1 Mk converges. Then

limm→∞

∞

∑k=m

Mk = 0

Proof: By assumption, there is N such that if m≥ N, then if n > m,∣∣∣∣∣ n

∑k=1

Mk−m

∑k=1

∣∣∣∣∣= n

∑k=m+1

Mk < ε/2

Then letting n→ ∞, one can pass to a limit and conclude that∞

∑k=m+1

Mk < ε

It follows that for m > N,∑∞k=m Mk < ε . The part about passing to a limit follows from the

fact that n→ ∑nk=m+1 Mk is an increasing sequence which is bounded above by ∑

∞k=1 Mk.

Therefore, it converges by completeness of R. ■

Theorem 13.8.3 For x ∈ S, if ∑∞k=1 |f k (x)|< ∞, then ∑

∞k=1f k (x) converges pointwise. If

there exists Mk such that Mk ≥ |f k (x)| for all x∈ S, then ∑∞k=1f k (x) converges uniformly.

Proof: Let m < n. Then∣∣∣∣∣ n

∑k=1

f k (x)−m

∑k=1

f k (x)

∣∣∣∣∣≤ ∞

∑k=m|f k (x)|< ε/2

whenever m is large enough due to the assumption that ∑∞k=1 |f k (x)|< ∞. Thus the partial

sums are a Cauchy sequence and so the series converges pointwise.If Mk ≥ |f k (x)| for all x ∈ S, then for M large enough,∣∣∣∣∣ n

∑k=1

f k (x)−m

∑k=1

f k (x)

∣∣∣∣∣≤ ∞

∑k=m|f k (x)| ≤

∞

∑k=m

Mk < ε/2

Thus, taking sup ∥∥∥∥∥ n

∑k=1

f k (·)−m

∑k=1

f k (·)

∥∥∥∥∥≤ ε/2 < ε

and so the partial sums are uniformly Cauchy sequence. Hence they converge uniformly towhat is defined as ∑

∞k=1f k (x) for x ∈ S. ■

Some of the following exercises have been essentially done in the above discussion.Try doing them yourself. There are also some new topics.

246 CHAPTER 13. SOME FUNDAMENTALS*Definition 13.8.1 The symbol Yj, f(a) means limy +. V7_, f(x) provided this limitexists. This is called pointwise convergence of the infinite sum. Thus the infinite sum meansthe limit of the sequence of partial sums. The infinite sum is said to converge uniformly ifthe sequence of paritial sums converges uniformly.Note how this theorem includes the case of 7, ax as a special case. Here the a; don’tdepend on x.The following theorem is very useful. It tells how to recognize that an infinite sum isconverging or converging uniformly. First is a little lemma which reviews standard calcu-lus.Lemma 13.8.2 Suppose M; > 0 and V_, My converges. Thenlim y My =0k=mm—s00Proof: By assumption, there is N such that if m > N, then ifn > m,n mYe Mi— Mik=l k=lThen letting n — o9, one can pass to a limit and conclude thaty. M<eék=m+1n= y? M, <e/2k=m+1It follows that form > N,Y7_,,, Mg < €. The part about passing to a limit follows from thefact that n + Yy_,,,.; Mx is an increasing sequence which is bounded above by Yi"; Mx.Therefore, it converges by completeness of R.Theorem 13.8.3 For x € S, if De) |f,(x)| < %, then Le, Ff, (ax) converges pointwise. Ifthere exists My such that My > |f ,(x)| for all x € S, then Ye_, f(x) converges uniformly.Proof: Let m <n. Then¥ fy(@)—Y f(a)k=1 k=1<¥ |f,(a)| <e/2k=mwhenever m is large enough due to the assumption that )7_, |; (a)| < ce. Thus the partialsums are a Cauchy sequence and so the series converges pointwise.If M, > |f,(x)| for all « € S, then for M large enough,¥ fea) —Y Fela) < ¥ |fe(@)|< Me <e/2k=1 k=1 k=m k=mThus, taking sup<e/2<eéYA O-E AOk=l k=land so the partial sums are uniformly Cauchy sequence. Hence they converge uniformly towhat is defined as )2_, f, (x) fora ¢ S. HiSome of the following exercises have been essentially done in the above discussion.Try doing them yourself. There are also some new topics.