# Is there any information on power series whose coefficients are only 0 and 1?

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

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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