Consider three real numbers a < c < b and a function f : (a, b)...

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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, denote the Taylor polynomial of order n of f around c i.e. n P₁(x) = f(k) (c) k! k=0 (x - c) k. Show that for any other polynomial Qn ‡ Pn of order n it holds f(x) - Pn(x) lim x+c 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. 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, denote the Taylor polynomial of order n of f around c i.e. n P₁(x) = f(k) (c) k! k=0 (x - c) k. Show that for any other polynomial Qn ‡ Pn of order n it holds f(x) - Pn(x) lim x+c 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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To prove the given statement we need to show that lim x c fx Pxfx Qx 0 where Px is the Taylor polyno... View the full answer