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18.100B.Homework12

# 18.100B.Homework12 - n is g n y = cy n for some constant...

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18.100B/C: Fall 2008 Homework 12 Available Wednesday, November 26 Due Friday, December 5 1. (a) (10pts) Problem 2, page 165 in Rudin . (b) (10pts) Problem 3, page 165 in Rudin . 2. (10pts) Show that f n ( x ) = q 1 n + x 2 converges uniformly on R to f ( x ) = | x | . 3. (15 pts) Problem 9, page 166 in Rudin . 4. (20pts) Problem 10, page 166 in Rudin . (Use some theorem from Rudin chapter 7 to prove Riemann-integrability.) 5. (15pts) Consider the exponential function g ( x ) = e x = k =0 1 k ! x k on R . Use Rudin Theorem 7.17 to reprove that g = g . (Hint: Let f n be the partial sums of e x . Note that there is an obvious choice for x 0 .) 6. Let f : R R be a smooth function (i.e. all derivatives exist) and fix x 0 R . (a) (5pts) Recall the Taylor series T ( x ) of f at x 0 , and write it as series of monomials T ( x ) = X n =0 g n ( x - x 0 ) . (A monomial of degree
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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 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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