Answer
$$1/2.$$
Work Step by Step
We have $$
\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\cdots\\
=\frac{1}{4}(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots)
.$$
This is a gemetric series with the sum
$$\frac{1/4}{1-(1/2)}=\frac{1}{4-2}=1/2.$$
