Answer
$4$
Work Step by Step
$\sum\limits_{k=1}^4(-1)^{k}cos {(k\cdot\pi)}=(-1)^{1}cos( {1\cdot\pi})+(-1)^{2}cos( {2\cdot\pi})+(-1)^{3}cos( {3\cdot\pi})+(-1)^{4}cos ({4\cdot\pi})=-cos( {1\cdot\pi})+cos( {2\cdot\pi})+-cos( {3\cdot\pi})+cos ({4\cdot\pi})=1+1+1+1=4$
