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A problem I've been working on has lead me to consider power series of the form $\sum_{n \geq 0} a_n x^n$ with $a_n = 0$ or $a_n = 1$ for all $n$.

Is there any literature available on these series? What do we know about them? I'm interested in convergence, analytic properties, and possibly even representation by known functions. Any help would be greatly appreciated.

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    J.M., $n$o, I'm cons$i$dering these series in general.2011-04-20

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If there's only a finite number of $a_j=1$, then, well, the series converges for all $x$.

If there's an infinite number of $a_j=1$ then it will converge for $-1 due to comparison with $\Sigma x^n = 1/(1-x)$ and diverge for $x=\pm 1$ due to terms not going to zero.

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For one thing, "representation by known functions" is clearly overly optimistic - there are uncountably many power series of this form, and only countably many can be explicitly named or described by any list of "known" functions.

Many examples of lacunary functions have 0/1 coefficients.

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    An infinite sequence of 0's and 1's also corresponds to a subset of the natural numbers (consider the set of indices where the sequence has the value 1). Thus, the set of such sequences is in 1:1 correspondence with the set of subsets of natural numbers, an uncountable set.2011-04-20