Answer
Divergent
Work Step by Step
We are given the geometric series:
$\sum_{k=1}^{\infty} \dfrac{1}{2}3^{k-1}$
Determine the elements of the geometric series:
$a_1=\dfrac{1}{2}\cdot 3^{1-1}=\dfrac{1}{2}$
$r=3$
We compute $|r|$:
$|r|=\left|3\right|=3$
Because $|r|=3>1$, the series diverges.