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Revision as of 19:14, 6 May 2005 editBradBeattie (talk | contribs)6,888 edits Made the proof a bit more brief← Previous edit Revision as of 19:16, 6 May 2005 edit undoMhowkins (talk | contribs)228 editsm →Explaination: minor spelling fixNext edit →
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== Explaination == == Explanation ==


The key step to understand here is that The key step to understand here is that

Revision as of 19:16, 6 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.

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

Explanation

The key step to understand here is that

∑ i = 0 ∞ ( 9 10 ) i = 1 1 − 9 10 {\displaystyle \sum _{i=0}^{\infty }\left({\frac {9}{10}}\right)^{i}={\frac {1}{1-{\frac {9}{10}}}}}

For further information, read up on geometric series and convergence.

External Proofs


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