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1: 2015-11-04 (水) 22:26:19 osinko  |
| | + | TITLE:実数の定義2 |
| | + | #jsmath |
| | + | ***メモ [#g1ddf4b8] |
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| | + | 境界が有理数の場合。0.4の切断 |
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| | + | \(0.3999...=0.3+0.0999...\\ =0.3+9\left( 0.01+0.001+0.0001+\cdots \right) \\ =0.3+9\left( { \left( \frac { 1 }{ 10 } \right) }^{ 2 }+{ \left( \frac { 1 }{ 10 } \right) }^{ 3 }+{ \left( \frac { 1 }{ 10 } \right) }^{ 4 }+\cdots \right) \) |
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| | + | \(\lim _{ n\rightarrow \infty }{ { S }_{ n }= } 9\left( { \left( \frac { 1 }{ 10 } \right) }^{ 2 }+{ \left( \frac { 1 }{ 10 } \right) }^{ 3 }+{ \left( \frac { 1 }{ 10 } \right) }^{ 4 }+\cdots { \left( \frac { 1 }{ 10 } \right) }^{ n } \right) =9\sum _{ k=3 }^{ \infty }{ { \left( \frac { 1 }{ 10 } \right) }^{ k-1 } } \\ =9\times \frac { \frac { 1 }{ 100 } \left( 1-{ \left( \frac { 1 }{ 10 } \right) }^{ n } \right) }{ 1-\frac { 1 }{ 10 } } =9\times \frac { \frac { 1 }{ 100 } \left( 1-0 \right) }{ 1-\frac { 1 }{ 10 } } =\frac { \frac { 9 }{ 100 } }{ \frac { 9 }{ 10 } } =\frac { 90 }{ 900 } =\frac { 1 }{ 10 } =0.1\) |
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| | + | \(=0.3+\lim _{ n\rightarrow \infty }{ { S }_{ n }= } 0.3+0.1=0.4\) |
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| | + | \(\therefore 切断\left( \frac { 3 }{ 10 } +9\sum _{ k=3 }^{ \infty }{ { \left( \frac { 1 }{ 10 } \right) }^{ k-1 } } \quad ,\quad \frac { 4 }{ 10 } \right) \) |
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