MIT18_100BF10_pset12

# MIT18_100BF10_pset12 - k →∞ P k x of the Taylor...

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18.100B : Fall 2010 : Section R2 Homework 12 Due Tuesday, November 30, 1pm Reading: Tue Nov.23 : sequences and series of functions, Rudin 7.1-17 Thu Nov.25 : holiday 1 . (a) Problem 2, page 165 in Rudin . (b) Problem 3, page 165 in Rudin . 2 . Consider the exponential function g ( x ) = e x = k =0 k 1 ! x k on R . Use Rudin Theorem 7.17 to prove that g = g . (Hint: Note that an unguarded lecturer essentially did this in class, so full and glorious details are expected.) 3 . Problem 9, page 166 in Rudin . 4 . Problem 10, page 166 in Rudin . (Use some theorem from Rudin chapter 7 to prove Riemann-integrability.) 5 . Let f : R R be a smooth function (i.e. all derivatives exist) and ﬁx x 0 R . The Taylor series T ( x ) of f at x 0 is deﬁned as the pointwise limit T ( x ) = lim

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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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MIT18_100BF10_pset12 - k →∞ P k x of the Taylor...

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