18.100B.Homework11

# 18.100B.Homework11 - f n a real valued sequence of...

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18.100B/C: Fall 2008 Homework 11 Available Wednesday, November 19 Due Wednesday, November 26 1. (20 pts) Problem 7, page 138 in Rudin . 2. (10 pts) Problem 8, page 138 in Rudin . 3. Use the two deﬁnitions given in the previous two problems to answer the following questions with a precise explanation: (a) (10 pts) For which α R does the integral Z 1 0 sin t t α dt converge (absolutely)? ( Hint : First try to understand the convergence of the integral of 1 t γ on the interval [0 , 1] ). (b) (10 pts) For which β R does the integral Z 1 e - t t β dt converge (absolutely)? ( Hint : First try to understand the convergence of the integral of 1 t γ on the interval [1 , ) ). 4. Suppose that
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Unformatted text preview: ( f n ) , a real valued sequence of functions, converges uniformly to f . (a) (8 pts) Prove that if g : R → R is a Lipshitz function, then the sequence g ( f n )) converges uniformly to g ( f ) . (b) (7 pts) Prove that if ( f n ) above are bounded functions, then f is bounded. 5. Let f n ( x ) = n c x (1-x 2 ) n , for all x ∈ R and n ∈ N . (a) (5 pts) Prove that ( f n ) converges pointwise in [0 , 1] for all c ∈ R . (b) (10 pts) Determine those c for which the convergence is also uniform in [0 , 1] . 6. (20 pts) Problem 4, page 165 in Rudin ....
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