How to approximate 1/3 by only add/subtracting powers of 2

Question

How approximate $\frac{1}{3}$ up to four significant digits by using only $\pm2^n$ where $n$ is a negative integer.

Preliminary attempt/example: $$0.33\approx0.25+0.0625=0.3125$$ $$=2^{-2}+2^{-4}$$

Answer 1

Hint: The series $$\frac12 - \frac1{2^2} + \frac1{2^3} - \cdots + \frac{(-1)^{n-1}}{2^n} + \cdots$$ converges to $\dfrac13$.

Answer 2

Following on from where you left off, the sequence $$2^{-2}+2^{-4}+2^{-6}+...$$ converges to $\frac 13$ using the formula for a geometric series