Is there a closed form for a finite sum $s_n=1+x+x^2+x^4+x^8+\ldots+x^{2^n}$? I apologize if this has been already discussed. Is there some literature on this I can take a look at?
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5There is none, otherwise an infinite sum of this kind would also fold nicely, but we know it does not. – Ivan Neretin Sep 19 '15 at 10:22
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It is more clear if you write $s_n=1+\sum_{k=0}^n x^{2^k}$. I wonder, however, why you've got the extra $1$ term. – celtschk Sep 19 '15 at 10:33
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@celtschk You're right. 1 is irrelevant :) – Poppy Sep 19 '15 at 10:34
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1@Did why do you think it's a duplicate? – Elaqqad Sep 19 '15 at 11:11
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1@Elaqqad Because the series in the radical is the same and because at least one of the answers explains what is known about the series itself. – Did Sep 19 '15 at 11:14
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See https://en.wikipedia.org/wiki/Lacunary_function. – lhf Sep 19 '15 at 11:31