Answer
Yes, the infinite geometric series has a limit, and the value of the limit is ${{S}_{\infty }}=\frac{320}{3}$.
Work Step by Step
$80+20+5+\cdots $
Here, ${{a}_{1}}=80$, ${{a}_{2}}=20$,
The value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{20}{80} \right| \\
& =\frac{1}{4}
\end{align}$
Thus, the series does have a limit.
Find the limit of the infinite geometric series for the formula ${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{80}{1-\frac{1}{4}} \\
& =\frac{80}{\frac{3}{4}} \\
& =\frac{4\cdot 80}{3} \\
& =\frac{320}{3}
\end{align}$
Therefore, the limit of the infinite geometric series is ${{S}_{\infty }}=\frac{320}{3}$.
