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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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    The term *lacunary function* will be helpful when you want to search for related topics, as mentioned by GEdgar.2012-10-27

1 Answers 1

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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!2012-11-28