Answer
The series is convergent with a sum $\frac{cos1}{1-cos1}$.
Work Step by Step
$\Sigma^{\infty}_{k=1} (cos 1)^{k} = \Sigma^{\infty}_{k=1} (cos 1)(cos 1)^{k-1}$
It is a geometric series with $r=cos1$ and $a=cos1$
Since $|r|=|cos1| \lt 1$, the series is convergent and has a sum
$\Sigma^{\infty}_{k=1} (cos 1)^{k} = \frac{a}{1-r}$
$= \frac{cos1}{1-cos1}$
$\approx 1.175343$
