Answer
$S_{\infty}=\frac{250}{3}\approx 83.33$
Work Step by Step
In order to determine whether the infinite geometric series has a limit we have to find $r$. If $|r| < 1$ the it has a limit.
Our geometric series has the terms:
$100, -20, 4,\cdots $.
We calculate the ratio:
$r=\frac{a_2}{a_1}=\frac{-20}{100}=-\frac{1}{5}$
Because $|r|=\left|-\frac{1}{5}\right|=\frac{1}{5}<1$, it follows that the series has a limit. Let's find it.
The limit of an infinite geometric series is
$S_{\infty}=\dfrac{a_1}{1-r}$.
In our case
$$S_{\infty}=\dfrac{100}{1-\left(-\frac{1}{5}\right)}=\dfrac{100}{\frac{6}{5}}=\frac{500}{6}=\frac{250}{3}\approx 83.33$$
