The sum in question is $$\sum_{k=0}^\infty x^{a^k}. $$ This comes from a more difficult summation, but if I can find the value of this one then the other one is solved.

Does anyone know the exact value of this? Any help in that direction would be really appreciated.

The original problem is in this post: A tricky infinite sum-- solution found numerically, need proof

$X Y^{\alpha} + X^2 Y^{\alpha + \alpha^2} +X^3 Y^{\alpha + \alpha^2 + \alpha^3} + ...=\sum\limits_{j = 1}^{\infty}X^j \prod\limits_{k = 1}^{j}Y^{\alpha^k}$ where $0 < \alpha < 1, 0 < X < 1$ and $Y>0$

  • $\begingroup$ I'm afraid not. All that's clear is that, in order for the sum to be convergent, $|x|<1$. Perhaps putting up the original question would be more useful ? $\endgroup$
    – Lucian
    Jan 3, 2014 at 17:30
  • $\begingroup$ It would definitely help if you put up the original question. $\endgroup$
    – user98602
    Feb 15, 2014 at 8:12
  • $\begingroup$ There it is. @Mike $\endgroup$ Feb 15, 2014 at 8:24
  • $\begingroup$ It looks as if my answer to math.stackexchange.com/questions/664438/… might match your problem. If it is so, and the approach meets your needs in principle then it might be adapted/refined if needed/if possible. $\endgroup$ Feb 17, 2014 at 17:33
  • $\begingroup$ Also you might google for "lacunary series". $\endgroup$ Feb 17, 2014 at 18:13

1 Answer 1


There seems to be no general closed expression for this. The problem is known with the keyword "lacunary series", and there is an informative wikipedia-page about this.


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