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.Would you mind showing me the detail, please. Thank you so much P₁ (x) = f(k) (c)k!Show that for any other polynomial Qnnlimx →Ck=0·(x – c)k.Pn of order n it holdsf(x) – Pn(x)f(x) - Qn(x)=0.Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of fvia a polynomial of order n.

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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

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 P_n denote the Taylor polynomial of order n of f around c i.e.

Would you mind showing me the detail, please. Thank you so much

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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