Mapping partial sums to [0,1]

Consider the binary number .abc... between 0 and 1 and take the ith term from the sum of some series if  the ith digit of the binary number is 1, These then get summed and then every possible partial sum gets mapped to a number between 0 and 1...

Considering the binary number .101101 and the series below...

The top would equal pi/4 going to infinity and the bottom uses the 1st, 3rd, 4th, and 6th term because there are 1's in the binary number in those places...

So any number even irrational or transcendental between 0 and 1 inclusively can be used to pick out a partial sum from a series, thus mapping all those infinite partial sums to the range [0, 1]