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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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Is there any pattern in the sequence $a_n$? – J. M. Apr 20 '11 at 2:27
J.M., no, I'm considering these series in general. – Antonio Vargas Apr 20 '11 at 2:34
up vote 7 down vote accepted

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 $-1<x<+1$ 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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Right, hence my question in the comments. If there's no pattern, the likelihood of a closed form is miniscule. – J. M. Apr 20 '11 at 2:45
I almost bit my fingers when I typed it, but I figured I should give it a shot anyway in the hopes that special cases have been studied. – Antonio Vargas Apr 20 '11 at 2:47
special cases have certainly been studied. Some of those series are rational functions, others are lacunary functions, etc. – Alon Amit Apr 20 '11 at 2:54
For a while I had trouble wrapping my head around the "uncountably many" in the above comment. But now I see that the series are in 1-1 correspondence with the real numbers between 0 and 1 by describing them with as binary decimals. I'll upvote this when I get my vote back later today. – Carl Brannen Apr 20 '11 at 16:32
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. – Alon Amit Apr 20 '11 at 19:24

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