f07HW3 - MAT 125A Homework 3 M.Fukuda Please submit your...

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Unformatted text preview: MAT 125A Homework 3 M.Fukuda Please submit your answers at the discussion session on October 25th 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, t R and f1 , f2 : (s, t) R. Take a (s, t) and, suppose limxa- f1 (x) and limxa- f2 (x) exist. Let Li = limxa- fi (x) for i = 1, 2. 1) Suppose L1 , L2 R. Then, show that L = limxa- f1 (x)+f2 (x) exits and L = L1 +L2 . 2) Suppose L1 R and L2 = . Then, does the limit L = limxa- f1 (x) + f2 (x) exist? Prove your idea. 3) Suppose L1 = - and L2 = . Then, does the limit L = limxa- f1 (x) + f2 (x) exist? Explain your idea briefly. 2. Find the redious of convergence and determine the exact interval of convergence. i) n x n n ; ii) n n3 3n xn ; iii) n 1 (n + 1)2 2n xn ; iv) n (-1)n n2 4n xn 3. Let fn (x) = x/n on [0, ). i) Find f (x) = lim fn (x). ii) Determine if fn f uniformly on [0, 1] and prove your idea. iii) Determine if fn f uniformly on [0, +) and prove your idea. 1 ...
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This note was uploaded on 04/08/2008 for the course MATH 125A taught by Professor Fukuda during the Fall '07 term at UC Davis.

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