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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 ( ² 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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- Fall '09