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18.100B.Homework9

18.100B.Homework9 - radius of convergence at any x ∈(0...

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18.100B/C: Fall 2008 Homework 9 Available Monday, November 3 Not Due If you would like feedback on your solutions, you can turn in the homework by 11am on Wednesday, November 12, in 2-108. 1. Show that sin( x ) ± x is a good approximation for small x by using Taylor’s theorem to obtain | sin( x ) - x | ≤ 1 6 | x | 3 x R . Use this to calculate the limit for different values of a R and c > 0 of the function x a sin( | x | - c ) (from Homework 8, Rudin pg.115 #13) as x → ∞ . 2. Use l’Hospital’s Rule to show that for any polynomial P ( x ) lim x →∞ P ( x ) e x = 0 and lim x →∞ ln( x ) P ( x ) = 0 . For the second limit (of course) assume that P ( x ) is not constant. 3. Show that the function f ( x ) = 1 x is analytic on (0 , ) by ﬁnding it’s Taylor series (and
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Unformatted text preview: radius of convergence) at any x ∈ (0 , ∞ ) . [ Hint: Rewrite 1 x into the form-β 1-β ( x-x ) and remember geometric series. ] 4. Consider the power series L ( z ) = ∑ ∞ n =1 (-1) n-1 n z n . a) What is its radius of convergence R ? b) For which z ∈ C does L ( z ) converge? c) On its domain of deﬁnition, calculate the derivative L ± ( z ) . d) Compare the result of c) with problem 3 and integrate (using your calculus knowl-edge) to show that L ( z ) = ln(1 + z ) . 5. Problem 1, page 196 in Rudin. 6. Problem 8, page 138 in Rudin. 7. Problem 3, page 138 in Rudin....
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