Unformatted text preview: n is g n ( y ) = cy n for some constant c .) (b) (5pts) For which r > do the Taylor polynomials P k ( x ) = ∑ k n =0 g n ( xx ) converge uniformly on B r ( x ) to T ( x ) ? (Hint: Compare r with the radius of convergence of T ( x ) . You can use e.g. Rudin Theorem 7.10.) (c) (10pts) Recall Taylor’s error formula for  f ( x )P k ( x )  . Deduce that f ( x ) = T ( x ) for all x ∈ B A ( x ) if A > satisﬁes A < lim n →∞ ˆ 1 n ! sup z ∈ B A ( x )  f ( n ) ( z )  !1 /n . 7. ( 15pts extra credit for those who like to ”ﬁll their stomach with a wriggly line” ) Problem 14, page 168 in Rudin ....
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 Fall '10
 Prof.KatrinWehrheim
 Exponential Function, Derivative, Taylor Series

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