Taylor Polynomial Lagrange Error Bound

Question

The function $$f(x)=\ln(1+x)$$ is approximated by its second-degree Taylor polynomial, $$T_2(x)$$, centered at $$x=0$$. Which of the following correctly states $$T_2(x)$$ and the associated Lagrange error bound?

Accepted Answer

$$T_2(x)= x-\frac{x^2}{2}$$ with error bound $$|R_2(x)| \le \frac{2|x|^3}{6(1+\xi)^3}$$, for some $$\xi$$ between 0 and $$x$$

Answer

$$T_2(x)= x-\frac{x^2}{2}$$ with no applicable error bound

Answer

$$T_2(x)= x-\frac{x^2}{2}+\frac{x^3}{3}$$ with error bound $$|R_3(x)| \le \frac{|x|^4}{24}$$

Answer

$$T_2(x)= x-\frac{x^2}{2}$$ with error bound $$|R_2(x)| \le \frac{|x|^3}{3!}$$

Taylor Polynomial Lagrange Error Bound

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