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Why 9.9999999...=10

Solution

Solution by Leonhard Euler: 
\[9,999 \ldots = 9+\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\cdots\]
\[=9\left(1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots\right)\]
\[=9 \frac{1}{1-\frac{1}{10}}\]
\[=9 \frac{10}{9}\]
\[=10\]
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