University of Florida/Egm6341/s11.team1.Gong/Mtg6

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Mtg 6: Sun, 16 Jan 11


Proof of Taylor series continued



Int. by parts (1)


Combine [ + β - α] into a single int. use (2) p.6-1 in (2) p.5-3:


HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3 more terms in Taylor series, i.e. ,

plus remainder

2)Use IMVT to expression remainder in terms of f(s)(ξ) for s belong[x0,x]

3) Assume(3)&(4)p.3-3 correct , do intergration by parts once more


to verify(3)&(4) p.3-3 for (n+1) expansionwith R(n+2)(x)

4) UseIMVT on(4) p.3-3to show(5)p.3-3


Use "g(x)" instead of "f(x)" to avoid confusion with "f(x)" is a Taylor Series.

HW*2.2: Constrast Taylor Series of f(.) around

for n = 0,1,2,...,10

Plot these series (for each n) Find (estimate) max


Note:Motivation for pf of Taylor series expansion. (similar technique will be used)

  • higher order analysis of Trap. rule (not in A.)
  • Richardson extrap.
  • clenshaw-Cwetis quadrature
  • chebyshew poly (orthog.) Recent devel. using chebyshew poly to solveL2_ODE_VC (Linear 2nd order ODE with varying coefficient ) combine of symbolics + numericsReference : Trefethen's chebfun.



Num. Int. Using Taylor Series cont'd

{\color{blue} \underset{f_{n}(x)=P_{n}(x)}{ \underbrace{{\color{black} \sum_{j=0}^{\n}\frac{x^{j-1}}{j!}}}{\color{black}dx}}}