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| JARGON! | |||
| 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. | |||
| == 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 following infinite geometric series is convergent: | |||
| :<math>\sum_{k=0}^\infty \left( \frac{1}{10} \right)^k = \frac{1}{1 - \frac{1}{10}}.</math> | |||
| ==Alternative proof== | |||
| A less mathematical proof goes as follows. Let ''x'' equal 0.999... Then, | |||
| :10''x''−''x'' = 9.999... − 0.999... | |||
| and so | |||
| :9''x'' = 9, | |||
| which implies that ''x'' = 1. | |||
| == See also == | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| == External proofs == | |||
| * | |||
| * | |||
| {{mathstub}} | |||
| ] | |||
Revision as of 17:05, 8 May 2005
JARGON!