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Can all algebraic numbers (i.e. quantities such as $3/5$, $\sqrt{2}$, $\sqrt{3}$, etc.) be expressed as an infinite sum whose summands never permanently vanish?

A well known example is $\sum_{n=1}^\infty\frac{1}{2^n}=1.$

I realise we could always multiple this infinite summation by any number we like, but what if this were not permitted?

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    Every real number, whether algebraic or otherwise can be represented thus: just use the decimal expansion, and chose the one which ends with all 9's from some point onwards if the decimal you chose happens to be a terminating one.2012-11-07

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