# How do I use a geometric series to prove that 0.999...=1?

Oct 4, 2014

$0.999 \ldots$

by separating digits,

$= 0.9 + 0.09 + 0.009 + \cdots$

by rewriting in the form of a geometric series,

$= \frac{9}{10} + \frac{9}{10} \left(\frac{1}{10}\right) + \frac{9}{10} {\left(\frac{1}{10}\right)}^{2} + \cdots$

by the sum formula for a convergent geometric series,

$= \frac{\frac{9}{10}}{1 - \frac{1}{10}} = 1$

I hope that this was helpful.