Answer
converges to $k$
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
Consider the series $\Sigma_{n=0}^{\infty} k(\dfrac{1}{2})^{n+1}$; which shows a convergent geometric series with $a=\dfrac{k}{2}$ and common ratio $r =\dfrac{1}{2}$
$S=\dfrac{a}{1-r}=\dfrac{\dfrac{k}{2}}{1-\dfrac{1}{2}}=k$
Hence, the series converges to $k$