hw4 - E . 6. Consider f ( x ) = 1 X n =1 1 1 + n 2 x : For...

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Homework 4 { Math 118B, Winter 2010 Due on Thursday, February 25th, 2010 1. Suppose ± increases monotonically on [ a;b ], g is continuous, and g ( x ) = G 0 ( x ) for a ± x ± b . Prove that Z b a ± ( x ) g ( x ) dx = G ( b ) ± ( b ) ² G ( a ) ± ( a ) ² Z b a Gd±: 2. Let ² 1 , ² 2 , ² 3 be curves in the complex plane, de±ned on [0 ; 2 ³ ] by ² 1 ( t ) = e it ; ² 2 ( t ) = e 2 it ; ² 3 ( t ) = e 2 ±it sin(1 =t ) : Show that these three curves have the same range, that ² 1 and ² 2 are recti±able, that the length of ² 1 is 2 ³ , that the length of ² 2 is 4 ³ , and that ² 3 is not recti±able. 3. Prove that every uniformly convergent sequence of bounded functions is uniformly bounded. 4. If f f n g and f g n g converge uniformly on E , prove that f f n + g n g con- verges uniformly. In, in addition, f f n g and f g n g are sequences of bounded functions, prove that f f n g n g converges uniformly on E . 5. Construct sequences f f n g and f g n g which converge uniformly on some set E , but such that f f n g n g does not converge uniformly on
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Unformatted text preview: E . 6. Consider f ( x ) = 1 X n =1 1 1 + n 2 x : For what values of x does the series converve absolutely? On what intervals does it converge uniformly? On what intervals does it fail to converge uniformly? Is f continuous wherever the series converges? Is f bounded? 7. Prove that the series 1 X n =1 ( 1) n x 2 + n n 2 converges uniformly in every bounded interval, but does not converge absolutely for any value of x . 1 2 8. Prove the following theorem, known as Mean Value Theorem for inte-grals: For a continuous function f ( x ) in the interval [ a;b ] there exists a value 2 [ a;b ] such that Z b a f ( x ) dx = f ( )( b a ) :...
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hw4 - E . 6. Consider f ( x ) = 1 X n =1 1 1 + n 2 x : For...

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