Unformatted text preview: radius of convergence) at any x ∈ (0 , ∞ ) . [ Hint: Rewrite 1 x into the formβ 1β ( xx ) and remember geometric series. ] 4. Consider the power series L ( z ) = ∑ ∞ n =1 (1) n1 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 knowledge) 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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 Fall '10
 Prof.KatrinWehrheim
 Calculus, Approximation, Rudin pg.115

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