APFIVE Taylor Polynomial Lagrange Error Bound
Which of the following is an example of calculating the Lagrange error bound for a Taylor polynomial approximation of $$f(x)=\ln(1+x)$$ near $$x=0$$?
A
$$T_2(x)= x-\frac{x^2}{2}$$ with no applicable error bound
B
$$T_2(x)= x-\frac{x^2}{2}$$ with error bound $$|R_2(x)| \le \frac{|x|^3}{3!}$$
C
$$T_2(x)= x-\frac{x^2}{2}+\frac{x^3}{3}$$ with error bound $$|R_3(x)| \le \frac{|x|^4}{24}$$
D
$$T_2(x)= x-\frac{x^2}{2}$$ with error bound $$|R_2(x)| \le \frac{2|x|^3}{6(1+\xi)^3}$$, for some $$\xi$$ between 0 and $$x$$
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