UCLA MATH 131A - homework6 (5 pages)

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homework6



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homework6

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Pages:
5
School:
University of California, Los Angeles
Course:
Math 131a - Analysis

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Math 131A Homework 6 There is no quiz for this homework but it will be examined on the final P 1 n n 1 Suppose n 0 an x is a power series and that an n 1 converges to a nonzero real number 1 Let lim an n and R 1 Prove that the power series converges for x R and diverges for x R P n Help Fix an x R and apply the root test to the series n 0 an x 2 Let f R 1 R be defined by f x 1 1 x a i Calculate the Taylor series for f x at 0 ii What is the radius of convergence R0 of this Taylor series iii Does the Taylor series converge to f on R0 R0 b i Use part a the substitution x 2y 1 and the identity 1 1 2 1 y 1 2y 1 to calculate a power series for f centered about 21 ii What is the radius of convergence of R1 2 of this power series iii Does the power series converge to f on 21 R1 2 12 R1 2 iv You just calculated a Taylor series without calculating any derivatives Use part i to read off the values of f n 12 for n N 0 3 Prove that there is an infinitely differentiable function g R R such that for all n N 0 g n 0 n2 Help Try to define g using a power series centered at 0 TURN OVER 1 4 a Use Taylor s theorem for f x sin x n 3 c 0 and x 1 to deduce that sin 1 0 b Use Taylor s theorem for f x sin x n 9 c 0 and x 4 to deduce that sin 4 0 c Use the intermediate value theorem to deduce that there is an x0 1 4 with sin x0 0 You have just proved that exists 5 Consider the inequality 2 exp x 1 x exp x for x 0 x2 a Prove it using the definition of exp x b Prove it using Taylor s theorem with n 2 for exp about 0 and the fact that exp is strictly increasing P 1 n 1 x2n 2 6 a Let f x What is f 0 What is x x3 f x n 1 2n 1 P 1 n 1 x2n 2 What is g 0 What is 1 x2 g x b Let g x n 1 2n c Calculate limx 0 x sin x 70 1 cos x105 Help Ponder x3 f x 70 x105 2 g x105 7 Prove that there is an infinitely differentiable function h R R such that for all n N 0 h n 0 n2 and such that h 6 g where g is the function you constructed in question 3 Help What should be true for h g Recall the rubbish Taylor series THE LAST TWO



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