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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 ( x-x ) 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 Taylors error formula for | f ( x )-P k ( x ) | . Deduce that f ( x ) = T ( x ) for all x B A ( x ) if A > satises A < lim n 1 n ! sup z B A ( x ) | f ( n ) ( z ) | !-1 /n . 7. ( 15pts extra credit for those who like to ll their stomach with a wriggly line ) Problem 14, page 168 in Rudin ....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
- Fall '10