1549901369-Elements_of_Real_Analysis__Denlinger_

9.2 Uniform Convergence 553

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Theorem 9.2.11 (Uniform Cauchy Criterion for Series) A series I: fk

k=O

of functions in F(S, JR) converges uniformly on S if and only if 'r:/c; > 0, 3

no EN 3 n > m :'.:'.: n 0 ==?- f: fk is bounded and II f: fkll < €.

k=m+I k=m+I

Proof. Exercise 13. •

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Corollary 9.2.12 If L fk = f uniformly on a set S, then llfkll ---+ 0.

k=O

Proof. Exercise 14. •

The following simple test is often useful in proving that a series of functions

is uniformly convergent.

Theorem 9.2.13 (Weierstrass M-Test) Let Un} be a sequence of functions

defined and bounded on a set S of real numbers. If there is a sequence of positive

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real numbers {Mk} such that I: Mk< oo and 'r:/k EN, llfkll :::; Mk on S, then

k=O

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the series L f k converges (absolutely and) uniformly on S.

k=O

Proof. Suppose Un}, S, and {Mk} are as described in the hypotheses.
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Since L Mk converges, it satisfies the Cauchy criterion for series (8.1.11), so
k=O
n
3 no E N 3 n > m :'.:'.: no ==> L Mk < c:. Thus,
k=m+I

n > m :'.:'.:no=?-II f: fkll :::; f: llfkll :::; f Mk< c:.

k=m+I k=m+I k=n+l

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By the uniform Cauchy criterion for series, L fk converges uniformly on S.

k=O

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Corollary 9.2.14 If L ak converges absolutely, then L ak sinbkx and

k=O k=O

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L akcosbkx converge uniformly on (-00,00) for any sequence {bk}.

k=O

Proof. Apply the Weierstrass M-test. •