Last modified on 27 November 2011, at 21:08

0.999.../Proof by the geometric series formula

AssumptionsEdit

ProofEdit

Using the series definition of the value of an infinite decimal,

0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots.\,

This is a geometric series with a common ratio of 1/10. Applying the geometric series formula,

0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,