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Is there a closed-form, in terms of elementary functions or otherwise, for the power series $x+x^2+x^4+x^8+x^{16}+...$, where each term is the square of the last?

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    It satisfies $p(x)=x+p(x^2)$ - not sure that helps much, though2012-10-27
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    Not an elementary function. But an example that is sometime given for a function with natural boundary on the whole unit circle.2012-10-27
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    The term *lacunary function* will be helpful when you want to search for related topics, as mentioned by GEdgar.2012-10-27

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