Chapter 10: Infinite Sequences and Series - Section 10.9 - Convergence of Taylor Series - Exercises 10.9 - Page 625: 38

$\lt 0.0026$

Recall the Taylor series for $\cos x=1-\dfrac{x^2}{2}+\dfrac{ x^3}{3}-....$ We can see that the term $1-\dfrac{x^2}{2}$ is too small by the Alternating Series Estimation Theorem. Therefore, the error is less than $\dfrac{(0.5)^4}{24} \lt 0.0026$ by the Alternating Series Estimation Theorem.