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n -f(x) (c) (x - c)*. Ph(x) = Σ k! k=0 Show that for any other polynomial Qn Pn of order n it holds f(x) - Pn(x) f(x) - Qn(x) lim x →C = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.

n -f(x) (c) (x - c)*. Ph(x) = Σ k! k=0 Show that for any other polynomial Qn Pn of order n it holds f(x) - Pn(x) f(x) - Qn(x) lim x →C = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider three real numbers a < c < b and a function f : (a, b) → R such that the (n + 1)-th derivative f(n+1) exists on (a,b) and is continuous. Let Pn denote the Taylor polynomial of order n of f around c  i.e.

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P₁ (x) = f(k) (c) k! Show that for any other polynomial Qn n lim x →C k=0 ·(x – c)k. Pn of order n it holds f(x) – Pn(x) f(x) - Qn(x) = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.
Transcribed Image Text:P₁ (x) = f(k) (c) k! Show that for any other polynomial Qn n lim x →C k=0 ·(x – c)k. Pn of order n it holds f(x) – Pn(x) f(x) - Qn(x) = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.
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