Chapter 10: Infinite Sequences and Series - Section 10.9 - Convergence of Taylor Series - Exercises 10.9 - Page 625: 36

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

Recall the Taylor series for $ e^x=1+x +\dfrac{x^2}{2}+\dfrac{ x^3}{6}-....$ We need to 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 $|R_4(0.5)| \leq (\sqrt {e}) \times \dfrac{|0.5-0|^{5}}{5!} \approx 7.03 \times 10^{-4}$ Thus, $\space Error \leq 7.03 \times 10^{-4}$