Chapter 9 - Power Series - 9.1 Approximating Functions with Polynomials - 9.1 Exercises - Page 673: 65

$2.67 \times 10^{-3}$

The remainder in the n-th order Taylor's polynomial $T_n(x)$ for a function $f(x)$ centered at $a$ can be expressed as: $|R_n(x)|=|f(x)-T_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!}$ We are given that $f(x)=\ln (1+x)$ and $T(x)=x-\dfrac{x^2}{2}$ and $n=2$ Thus, we have $|R_n(x)|\leq 2 \times \dfrac{|0.2-0|^{2+1}}{(2+1)!} \approx 2.67 \times 10^{-3}$