hw9-555-2011 - f ( x ) = x . (4) Let f n ( x ) = x n e-nx ....

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Homework 9. (1) Let f n ( x ) = x 2 n 1+ x 2 n . Prove that f ( x ) = lim n →∞ f n ( x ) exists for all x R . Does ( f n ) converge uniformly to f ? (2) Define f n : [0 , 1] [0 , 1] by f n ( x ) = x n (1 - x ). Prove that f n converges uniformly to 0. (3) Prove that f n ( x ) = nx + sin( nx 2 ) n converges uniformly to f on [0 , 1], where
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Unformatted text preview: f ( x ) = x . (4) Let f n ( x ) = x n e-nx . Prove that f n converges uniformly (Hint: Use the Weierstrass M-test). (5) Prove that n =1 nx 2 n 3 + x 3 converges uniformly on [0 , 2]. 1...
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