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Why 0.9999999...=1

Solution

The solution is proposed by the mathematician Leonhard Euler. Let’s start

\\[0,999 \ldots =\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\cdots\]

\[=9\left(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots\right)\]

\[=9 \frac{\frac{1}{10}}{1-\frac{1}{10}}\]

\[=9 \frac{1}{9}\]

\[=1\]

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