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Unformatted text preview: k →∞ P k ( x ) of the Taylor polynomials k ± f ( n ) ( x ) P k ( x ) = ( x − x ) n . n ! n =0 (a) For which r > do the Taylor polynomials P k converge uniformly on B r ( x ) to T ? (Hint: Write T ( x ) as a series and compare r with its radius of convergence. You can use e.g. Rudin Theorem 7.10.) (b) Recall Taylor’s error formula for  f ( x ) − P k ( x )  . Deduce that f = T on the ball B A ( x ) if A > satisﬁes ² ³ − 1 /n 1 A < lim sup  f ( n ) ( z )  . n →∞ n ! z ∈ B A ( x ) MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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 Fall '10
 Prof.KatrinWehrheim
 Exponential Function, Derivative, Taylor Series, Sequences And Series

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