Half of a Series

The square can be divided into L-shaped regions; one such L-shaped region is shown below, on the left. A blue square takes up exactly 1/3 of each L-shaped region, so all the blue squares add up to 1/3 of the square.

https://latex.artofproblemsolving.com/f/0/b/f0b0d378c16007488639dde8c3db62d0b4b19757.png

(Image courtesy of https://artofproblemsolving.com/wiki/index.php/Proofs without words .)

For every white double-square, there is a blue square. Hence, 1/3.

Relevant wiki: Geometric Progression Sum

The blue area:

$\frac {1}{4} + \left ( \frac {1}{4}\right)^2 + \left( \frac {1}{4} \right)^3 + \cdots= \frac {1}{4} \times \frac {1}{ 1 -\frac {1}{4} } = \frac {1}{4} \times \frac {4}{3} = \boxed { \frac {1}{3} }$

Let the area of blue region be 'x'.

Therefore, x=1/4 + 1/16 +1/64...........

Taking 1/4 common

x= 1/4(1+ (1/4 + 1/16 + 1/64....))

Therefore,x=1/4(1+x)

4x = 1+x

By solving we get x=1/3

The blue area equals: $\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+\cdots$ Which can be reached drawing the same square, but dividing like Realizing that the shaded area equals $\frac{1}{3}$

The color part is 1/4 + 1/16 + 1/64 + .... = sum (1/4)^I. This is a geometric series, then use the formua, we can have 1/3 since 1/4 is less than 1.