Answer
No, the infinite geometric series does not have a limit.
Work Step by Step
The provided series is $4-6+9-\frac{27}{2}+\cdots $.
Here, ${{a}_{1}}=4$, $n=\infty $ and
$\begin{align}
& r=\frac{-6}{4} \\
& =-\frac{3}{2}
\end{align}$
Since $\left| r \right|=\frac{3}{2}$ is not less than 1, ${{S}_{\infty }}$doesn’t exist.
Thus, the infinite geometric series $3+15+75+\cdots $ does not have a limit.
