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

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

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    Note that $\frac13 = \frac{1}{1+2}$. Have you heard about geometric series? – 2017-01-13
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    You should convert it into binary system. – 2017-01-13
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    If you want positive terms, you can use binary representation. – 2017-01-13
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    Sorry guys, that was a bit of a dumb question in hindsight – 2017-01-13

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

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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