1549901369-Elements_of_Real_Analysis__Denlinger_

554 Chapter 9 • Sequences and Series of Functions

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Corollary 9.2.15 A power series I: ak(x - c)k converges uniformly (and ab-

k=O

solutely) in any compact interval [c - r , c + r], where 0 < r < p and p is the

radius of convergence of the series.

Proof. Exercise 15. •

Thus, power series converge both absolutely and uniformly in any interval

in the interior of the interval of convergence. But for series of functions in

general, absolute and uniform convergence do not necessarily go together. For

an example of a series of functions that converges absolutely but not uniformly,

see Exercise 16. For a power series that converges uniformly but not absolutely

on an interval, see Exercise 17.

Many of the facts we have proved about series of real numbers carry over

to uniformly convergent series of functions. One such example is the following.

Theorem 9.2. 16 (Dirichlet's Test for Uniform Convergence of Se-

ries) Suppose {fk} and {gk} are sequences of functions in :F(S, IR) such that

n

(a) the sequence {Sn} of partial sums Sn= I: fk is uniformly bounded;

k=l

(b) '<Ix E S , the sequence {gk ( x)} is monotone decreasing and nonnegative;

( c) gk ___.. 0 uniformly on S.

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Then I: fkgk converges uniformly on S.

k=l

Proof. Exercise 1 8. •

.

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sin kx

Example 9.2.17 The senes L -k-

k=l

(a) converges uniformly on [<5, n - <5] for any 0 < 5 < i;

(b) converges pointwise, but not uniformly, on [O, n].

7f

Proof. (a) Let 0 < 5 <

2

. We shall apply Dirichlet's test (9.2.16) with
fk(x) = sinkx and gk(x) = t· Let x E [<5,n - <5]. As shown in Lemma 8.5.5,

I

L n sin kx I :s; -^1 1

1

-. - x-

1

:s; -. - 0.
k=l sm 2 sm 2
n
Hence the partial sums I: sin kx are uniformly bounded on [ 5, n - <5].
k=l
The sequence {gk} = { t} is monotone decreasing and nonnegative, and