Chapter 9 - Power Series - 9.1 Approximating Functions with Polynomials - 9.1 Exercises - Page 673: 66

$2.5 \times 10^{-3}$

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 $f(x)=\sqrt (1+x)$ and $T(x)=1+\dfrac{x}{2}$ and $n=1$ Thus, we have $|R_n(x)|\leq 0.5 \times \dfrac{|0.2-0|^{1+1}}{(1+1)!} \approx 2.5 \times 10^{-3}$