Answer
Yes, the infinite geometric series has a limit and the value of the limit is ${{S}_{\infty }}=27$.
Work Step by Step
$18+6+2+\cdots $
Here, ${{a}_{1}}=18$,${{a}_{2}}=6$
The value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{6}{18} \right| \\
& =\frac{1}{3}
\end{align}$
Thus, the series does have a limit.
Find the limit of the infinite geometry series for the formula ${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{18}{1-\frac{1}{3}} \\
& =\frac{18}{\frac{2}{3}} \\
& =\frac{3\cdot 18}{2} \\
& =27
\end{align}$
Therefore, the limit of the infinite geometric series is ${{S}_{\infty }}=27$.
