Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
3  
It satisfies $p(x)=x+p(x^2)$ - not sure that helps much, though –  Mark Bennet Oct 27 '12 at 16:21
3  
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
3  
The term lacunary function will be helpful when you want to search for related topics, as mentioned by GEdgar. –  sos440 Oct 27 '12 at 17:55
add comment

1 Answer 1

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."

share|improve this answer
    
Helpful references, Argon. BTW, hello! –  amWhy Nov 28 '12 at 1:41
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.