Is the sum of series $\displaystyle \sum_{n=0}^\infty \frac1{2^{2^n}} = \frac12 + \frac14 + \frac1{16} + \frac1{256} + \frac1{65536} + \dotsb \approx 0.8164215090218931$ representable in a closed form? If so, what it is?
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$\begingroup$ Related MSE question on $\sum_k x^{2^k}$: math.stackexchange.com/q/404510/163 $\endgroup$– Tobias KienzlerCommented Oct 25 at 2:00
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If you allow for a named number to be a closed form representation, the answer is "yes".
$\sum_{n=0}^\infty (1/2)^{2^n}$ is known as the Kempner number [1], a transcendental number [2].
More generally, numbers of the type $\sum_{n=0}^\infty x^{2^n}$ with algebraic $x\in\mathbb{C}$, $|x|<1$, are known to be transcendental, as first proven by Mahler [3].
[1] A.J. Kempner, On Transcendental Numbers (1916).
[2] Boris Adamczewski, The many faces of the Kempner number (2013).
[3] K. Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen (1930).
answered Jul 16 at 10:06
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2$\begingroup$ Since 2 is rational, doesn't this also just follow from Liouville's theorem? Mahler's theorem is interesting precisely because it includes irrational algebraic numbers. $\endgroup$– JoshuaZCommented Jul 16 at 11:30
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1$\begingroup$ if I understand Adamczewski correctly, Liouville needs a faster than exponential growth of the exponent (so $n!$ rather than $2^n$) $\endgroup$ Commented Jul 16 at 11:37
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2$\begingroup$ Huh. For some reason I thought exponential was sufficient. $\endgroup$– JoshuaZCommented Jul 16 at 14:59
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$\begingroup$ Here's a scan of Mahler's article: gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002372460 $\endgroup$ Commented Oct 25 at 1:59