| Revision as of 15:47, 7 May 2005 editBradBeattie (talk | contribs)6,888 edits →Explanation: Added see also← Previous edit | Revision as of 18:49, 7 May 2005 edit undo213.216.199.18 (talk)No edit summaryNext edit → | ||
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| This is crab | |||
| 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 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
This is crab