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? 

Hint:use  Maclaurin series expansion of exponential function 
$${ e }^{ ix }=1+ix-\frac { x^{ 2 } }{ 2! } -\frac { i{ x }^{ 3 } }{ 3! } +...+\frac { { \left( ix \right)  }^{ k } }{ k! } +\frac { { \left( ix \right)  }^{ k+1 } }{ \left( k+1 \right) ! } $$
