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A Liouville number is a number which can be approximated very closely be a sequence of rational numbers (here is the rigorous definition I am working off of: http://en.wikipedia.org/wiki/Liouville_number).

I'm looking for an example of a Liouville number which cannot be approximated by a sequence of rational numbers with a denominators which are all a constant c multiplied by powers of some number a.

For instance, the Louiville constant ($0.110001000000000000000001$...) can be approximated by the sequence $\frac{1}{10}$, $\frac{11}{10^2}$, $\frac{110001}{10^6}$, etc, which is not what I am looking for because in each case the denominator is a power of $10$. In this case, we would say that $c=1$, $a=10$, and the denominator is always of the form $c \cdot a^n$ for some positive $n$.

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    It is not clear to me what you mean. Do you mean you are looking for a Liouville number which does not satisfy the condition in the Wikipedia article for rational numbers p/q such that q is always c times a power of a?2010-08-17
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    How about $1 + 1/2^{2!} + 1/3^{3!} + 1/4^{4!} + \ldots$?2010-08-17
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    Personally I think anon's comment should be an answer.2011-12-31
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    Except that proving it's Liouville might not be so simple. However, modifications can be made...2012-01-01
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    Proving that anon's series is a Liouville number is not hard: the tail starting at the $(n+1)$st term is at most $2/(n+1)^{(n+1)!}$, while the denominator of the sum of the first $n$ terms is at most $n^{2n!}$. Therefore the truncation at the $n$th term gives a rational approximation of the form $|c - p/q| < 1/q^{n+1}$ (where $c$ is the sum of the infinite series).2012-01-01

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