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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, though – Mark Bennet Oct 27 '12 at 16:21
Not an elementary function. But an example that is sometime given for a function with natural boundary on the whole unit circle. – GEdgar Oct 27 '12 at 16:50
The term lacunary function will be helpful when you want to search for related topics, as mentioned by GEdgar. – Sangchul Lee Oct 27 '12 at 17:55

The series

$$\sum_{n=0}^\infty x^{\large-2^n}$$

generally does not have a closed form. This is just your series where $x \mapsto \dfrac{1}{x}$.

When $2 \le x \le 10$, the decimal expansion is given by the OEIS. When $x=2$, the number is called the "Kempner-Mahler number." The case when $x=10$ seems to be called the "Fredholm-Rueppel Sequence" and has many other interesting properties.

It has also been shown that the number, $M$, generated by the sum $x=2$ is transcendental by Mahler, and Knight showed that this was true for all $x\ge 2$. (Summarized here)

The continued fraction for this series is discussed for $x \ge 3$ in J. Shallit's "Simple continued fractions for some irrational numbers."

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Helpful references, Argon. BTW, hello! – amWhy Nov 28 '12 at 1:41

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