Answer
The series is divergent.
See graph
Work Step by Step
$a_{n}=\Sigma^{\infty}_{n=1}cosn$
partial sum: $s_{n}=a_{1} + a_{2}+... +a_{n}$
$n=1$ $s_{1}=0.5403$ $a_{1}=0.5403$
$n=2$ $s_{2}=0.1242$ $a_{2}=-0.4161$
$n=3$ $s_{3}=-0.8658$ $a_{3}=-0.9900$
$n=4$ $s_{4}=-1.5195$ $a_{4}=-0.6536$
$n=5$ $s_{5}=-1.2358$ $a_{5}=0.2837$
$n=6$ $s_{6}= -0.2756$ $a_{6}=0.9602$
$n=7$ $s_{7}=0.4783$ $a_{7}=0.7539$
$n=8$ $s_{8}=0.3328$ $a_{8}=-0.1455$
$n=9$ $s_{9}=-0.5784$ $a_{9} =-0.9111$
$n=10$ $s_{10}=-1.4174$ $a_{10}=-0.8391$
The series appears to be divergent because it oscillates from negative to positive forever.