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We know that the extended Euclidean division of 8 by 3 results in the periodic decimal number 2,666 ... We can also see that the series indicated by the decimal number, this is, 2 + 6/10 + 6/100 + 6/1000 + ... converges to 8/3. Let a and b be positive integers, with a> b. Suppose that the extended Euclidean division of a by b results in a periodic decimal number (simple or compound). How can we prove that the series indicated by this decimal number converges to a / b? I need help.

Paulo Argolo
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    I don't know where you're stuck, but if I had to guess it's this: With, for instance, $\frac2{11} = 0.18181818\ldots$, don't use $1/10 + 8/100 + 1/1000 + 8/10,000 + \cdots$, but use $18/100 + 18/10,000 + 18/1000,000 + \cdots$ instead. – Arthur Aug 08 '17 at 11:53
  • My difficulty lies in generalization. – Paulo Argolo Aug 08 '17 at 13:35

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