UCLA MATH 131A - homework6 (5 pages)

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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 ARE FOR THE KEEN 8 Calculate f 0 0 f 00 0 and f 000 0 where f x sin x cos x 1808 exp x601 1 3x3 5x4 2754x232 P n Help If you can figure out a1 a2 a3 in a power series expansion n 0 an x for f x then 0 00 000 you are basically done f 0 f 0 and f 0 are given by a1 2a2 and 6a3 How can you do this Multiply like polynomials there are a lot of terms you can ignore 9 Prove that there is an infinitely differentiable function f R R such that f 0 f 0 0 0 1 f 00 0 96 and x2 f 00 x xf 0 x x2 4 f x 0 for all x R Help Try to write f as a power series centered at 0 The function you have constructed is called the second Bessel function 2 Math 131A Homework 6 Solutions 1 1 1 Fix x R Then limn an xn n x limn an n x x R Thus the root P x x n test tells us that n 0 an x converges when R 1 and diverges when R 1 P n 2 a i n 0 x ii 1 iii Yes because we know about geometric series b i X X 1 1 n 1 n n 1 y 2 2 2y 1 2 1 y 1 2y 1 2 n 0 n 0 ii 12 iii Yes iv f n 21 n 2n 1 P n2 n 3 Let g x n 0 n x This has infinite radius of convergence since 0 It is infinitely differentiable with the correct derivatives at 0 by the off syllabus theorem 4 a Taylor s theorem tells us that there exists a y 0 1 with sin 1 1 Since cos y 1 sin 1 5 6 cos y 3 0 b Taylor s theorem tells us that there exists a y 0 4 with sin 4 4 43 45 47 49 cos y 3 5 7 9 Since cos y 1 sin 4 4 43 45 47 49 0 3 5 7 9 c By the intermediate value theorem we deduce that there is an x0 1 4 with sin x0 0 5 a We note that 2 exp x 1 x 2 X xn X 2xn 2 X 2xn 2 x2 x n n n 2 n 2 n 2 The result follows from the fact that for all n N 0 2 1 n 2 n and the fact that this inequality is strict when n 1 3 n 0 b Let x 0 Taylor s theorem says that there is a y 0 x such that exp x 1 x Thus 2 exp x 1 x x2 exp y 2 x 2 exp y exp x since exp is strictly increasing 6 a f 0 16 x x3 f x sin x b g 0 12 1 x2 g x cos x c The off syllabus theorem says that f and g are continuous so h x Thus f x 70 g x105 is continuous x sin x 70 x3 f x 70 f x 70 f 0 70 2 lim lim 70 105 105 2 105 105 x 0 1 cos x x 0 x g x x 0 g x g 0 6 lim 7 Let h f g where f is the function with the rubbish Taylor series and g is the function of question 3 For all n N 0 we have h n 0 f n 0 g n 0 0 n2 n2 Since f is not the zero function h 6 g 8 We consider the Taylor series of f x but ignore terms xn with n 4 x2 x2 1808 x2 1808x2 x 1 1 x 1 1 3 2 3 2 P n 0 00 3 We have f x x 3 1 1808 n 4 an x so that f 0 1 f 0 0 and 2 x 1 1808 000 f 0 3 1 3 1808 5425 3 2 9 Let f …


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