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In ], one could easily fall in the trap of thinking that while 0.999... is certainly close to 1, nevertheless the two are not equal. Here's a proof that they actually are.
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== Proof ==

{|-
|<math>0.999\ldots</math>
|<math>= \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots</math>
|-
|
|<math>= -9 + \frac{9}{1} + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots</math>
|-
|
|<math>= -9 + 9 \times \sum_{k=0}^\infty \left( \frac{1}{10} \right)^k</math>
|-
|
|<math>= -9 + 9 \times \frac{1}{1-\frac{1}{10}}</math>
|-
|
|<math>= 1.\,</math>
|}

== Explanation ==

The key step to understand here is that the infinite geometric series is convergent.
:<math>\sum_{k=0}^\infty \left( \frac{1}{10} \right)^k = \frac{1}{1 - \frac{1}{10}}.</math>

== See also ==

* ]
* ]
* ]
* ]

== External proofs ==

*
*
*

{{mathstub}}
]

Revision as of 18:49, 7 May 2005

In mathematics, one could easily fall in the trap of thinking that while 0.999... is certainly close to 1, nevertheless the two are not equal. Here's a proof that they actually are.

Proof

0.999 … {\displaystyle 0.999\ldots } = 9 10 + 9 100 + 9 1000 + ⋯ {\displaystyle ={\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots }
= − 9 + 9 1 + 9 10 + 9 100 + 9 1000 + ⋯ {\displaystyle =-9+{\frac {9}{1}}+{\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots }
= − 9 + 9 × ∑ k = 0 ∞ ( 1 10 ) k {\displaystyle =-9+9\times \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}}
= − 9 + 9 × 1 1 − 1 10 {\displaystyle =-9+9\times {\frac {1}{1-{\frac {1}{10}}}}}
= 1. {\displaystyle =1.\,}

Explanation

The key step to understand here is that the infinite geometric series is convergent.

∑ k = 0 ∞ ( 1 10 ) k = 1 1 − 1 10 . {\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}={\frac {1}{1-{\frac {1}{10}}}}.}

See also

External proofs

Template:Mathstub

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