I want to show that $$ \left |e^{ix} - \sum_{k=0}^n\frac{(ix)^k}{k!}\right | \le \min\left \{\frac{|x|^{n+1}}{(n+1)!}, \frac{2|x|^n}{n!} \right\} $$
The left term in the minimum is just an estimated Lagrange remainder, but I don't know how to get the second one.