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Show that the decimal number $0.12624120720...$ obtained by concatenating the digits of $n!$ successively with $n = 1, 2, 3, ...$ represents an irrational number.

A rational number either has a terminating decimal expansion or an eventually repeating decimal expansion. $0.12624120720... $ is clearly not a terminating sequence. $n!$ is unique for each $n \in \mathbb N$ But how do we prove that uniqueness means non-repeating here?

Gauri Sharma
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  • That was asked very recently: https://math.stackexchange.com/questions/2527848/show-that-0-12624120720-is-irrational –  Nov 20 '17 at 17:51

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Hint: can you prove that there are arbitrarily long sequences of zeroes - eventually longer than any claimed finite period?

Mark Bennet
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  • I have shown that the highest power of $10$ dividing $N!$ is the highest power of $5$ dividing $N!$. How do I go further? – Gauri Sharma Nov 21 '17 at 00:34