Chapter 9 - Section 9.9 - Convergence of Taylor Series - Exercises - Page 542: 36

Error $\leq 7.03 \times 10^{-4}$

Taylor series for $ e^x $ can be defined as: $ e^x=1+x +\dfrac{x^2}{2}+\dfrac{ x^3}{6}-....$ Consider the Remainder Estimation Theorem to find $|f^{5} | \leq M $. So, $|R_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!}$ and $|R_4(0.5)| \leq (e^{1/2}) \times \dfrac{|0.5-0|^{5}}{5!} \approx 7.03 \times 10^{-4}$ So, Error $\leq 7.03 \times 10^{-4}$