f07HW5 - , a ] for any a (0 , 1). 3) Does the series...

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MAT 125A Homework 5 M.Fukuda Please submit your answers at the discussion session on November 8th Thursday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Let S R and f, f n : S R for n N . Suppose f n f uniformly on S . Then, show that { f n } n is uniformly Cauchy on S . 2. Consider s n =1 x n 1 + x n . 1) Show that the series converges for x [0 , 1). 2) Show that the series converges uniformly on [0
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Unformatted text preview: , a ] for any a (0 , 1). 3) Does the series converge uniformly on [0 , 1)? Prove your idea. 3. Let s ( x ) = s n =0 (-1) n x 2 n +1 (2 n + 1)! c ( x ) = s n =0 (-1) n x 2 n (2 n )! . Here, s ( x ) = sin( x ) and c ( x ) = cos x but without using these facts answer the following questions. 1) Show that s = c and c =-s . 2) Show that ( s 2 + c 2 ) = 0. 3) Show that s 2 + c 2 = 1. 1...
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