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LSpice
LSpice
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Any number with of a form $\frac{1}{n}$ finite repetitive serieshas a decimal with a repetend of decimalsfinite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}$=0.929292...$\frac{92}{99}=0.929292\ldots$ in which case it is 92 that is repeating and the length of the series is 2.) but never longer than n (Provable by Dirichlet principle). Is there a way to find ALL numbers of the form $\frac{1}{n}$ which have repetitive seriesrepetends of length EXACTLY n, are$n$? Are there infinitely many of them and does the $\sum_{n \thinspace with \thinspace this \thinspace property}\frac{1}{n}$$\sum_{\text{$n$ with this property}}\frac{1}{n}$ converge or not?

Any number with of a form $\frac{1}{n}$ finite repetitive series of decimals (Example: $\frac{92}{99}$=0.929292... in which case it is 92 that is repeating and the length of the series is 2.) but never longer than n (Provable by Dirichlet principle). Is there a way to find ALL numbers of form $\frac{1}{n}$ which have repetitive series of length EXACTLY n, are there infinitely many of them and does the $\sum_{n \thinspace with \thinspace this \thinspace property}\frac{1}{n}$ converge or not?

Any number with of a form $\frac{1}{n}$ has a decimal with a repetend of finite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}=0.929292\ldots$ in which case it is 92 that is repeating and the length of the series is 2.) Is there a way to find ALL numbers of the form $\frac{1}{n}$ which have repetends of length EXACTLY $n$? Are there infinitely many of them and does $\sum_{\text{$n$ with this property}}\frac{1}{n}$ converge or not?

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Vuk Jovovic
Vuk Jovovic
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Is there a way to specify a special kind of reciprocals of natural numbers?

Any number with of a form $\frac{1}{n}$ finite repetitive series of decimals (Example: $\frac{92}{99}$=0.929292... in which case it is 92 that is repeating and the length of the series is 2.) but never longer than n (Provable by Dirichlet principle). Is there a way to find ALL numbers of form $\frac{1}{n}$ which have repetitive series of length EXACTLY n, are there infinitely many of them and does the $\sum_{n \thinspace with \thinspace this \thinspace property}\frac{1}{n}$ converge or not?