Ivan Shishkin, Rye (1878)

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0.999999...=10.999999...=1

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Solution

Solution by Uettechat · FR

0.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(101+102+103+...)=9n110k=910111/10=10.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(10^{-1}+10^{-2}+10^{-3}+...)=9\sum_{n\geq1}10^{-k}=\frac{9}{10}\frac{1}{1-1/10}=1

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