What is the name of a sequence in which each term is $r^s$ where $r$ is some constant and s is variable changing exponentially? How would one sum up such a thing?
An example would be:
$1 + (1/2)+ (1/2)^2 + (1/2)^4 + (1/2)^8 +\ ...$
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5A variable changing geometrically, huh? – Alexander Gruber Feb 25 '14 at 06:05
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Yeah, it's kind of a geometric series, but it doesn't sum well. – Jackson Feb 25 '14 at 06:13
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1So in your example you want to look at the series $\Sigma 2^{-2^n}$? Do you mean series where $s$ increases exponentially? – Alexander Gruber Feb 25 '14 at 06:23
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@AlexanderGruber Yep, pretty much. I'm still learning about this stuff, sorry if it's not too complicated. – Jackson Feb 25 '14 at 06:25
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@AlexanderGruber Oh whoops. let me change it in my question. thank you. – Jackson Feb 25 '14 at 06:27
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Power series such as $$\sum_{k=0}^\infty z^{2^k}$$ which have exponentially large gaps between the powers of successive terms are called lacunary functions. I don't think there is any closed-form expression for such a series. My understanding is that they have very strange behavior as $|z| \rightarrow 1$ for complex numbers $z$.
Jair Taylor
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