1/4 + 1/16 + 1/64 + 1/256 + ⋯

Neat. Of course, one easy way to see it is analogous to the base 10

If x = 0.3333... then 10 x = 3.3333... so 9x = 3 and thus x = 3/9 = 1/3.

but in binary.

In binary, clearly the sum would be represented as

0.01010101...

so if x = 0.01010101... then 100 x = 1.01010101... so 11 x = 1 and thus x = 1 / 11, or in decimal 1/3.