Prove $|e^{ix}-1-ix-\cdots-\frac{(ix)^k}{k!}| \leq \frac{|x|^{k+1}}{(k+1)!}$ I would like to prove 
$|e^{ix}-1-ix-\cdots-\frac{(ix)^k}{k!}| \leq \frac{|x|^{k+1}}{(k+1)!}$, which trick can be used? 

We need to put conditions on $x$. For example, if $x=-i$, then the absolute value of the difference on the left is $\frac{1}{(k+1)!}+\frac{1}{(k+2)!}+\cdots$.
