How to approximate 1/3 by only add/subtracting powers of 2 How approximate $\frac{1}{3}$ up to four significant digits by using only $\pm2^n$ where $n$ is a negative integer.
Preliminary attempt/example:
$$0.33\approx0.25+0.0625=0.3125$$
$$=2^{-2}+2^{-4}$$

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