A ride to infinity.....

If we notice clearly, this is a geometric sequence with the common ratio as $\frac{1}{4} = r$ . Therefore since $r$ is between $-1$ and $1$ , the sum to infinity exists. Using the formula of sum to infinity:

$\large \color{#3D99F6} \frac{a}{1 - r} \text{where 'a' is the first term}$

So we can substitute the values in our equation:

$\large \color{magenta}= \frac{0.5}{1 - 0.25} \\ \\ \large \color{magenta}= \frac{0.5}{0.75} \\ \\ \large \color{#EC7300} \boxed {= \frac{2}{3}}$

$\begin{aligned} S &= \frac 12 + \frac 18 + \frac 1{32} + \frac 1{128} + \cdots \\ \frac 14 S &= \frac 18 + \frac 1{32} + \frac 1{128} + \frac 1{512} + \cdots \\ \frac 34 S &= \frac 12 \\ \implies S &= \boxed{\dfrac 23} \end{aligned}$