How to sum the series:
$$\sum _{ n=0 }^{ n=\infty }{ \frac { 1 }{ { 2 }^{ { 2 }^{ n } } } }$$
PS: Just a hint would suffice.
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2$\begingroup$ There doesn't seem to be a closed form for this. The decimal expansion of it is OEIS A007404 and there are not that many references for this number. If you just want a number, the OEIS page has a link to the first 20000 digits of it. $\endgroup$– achille huiMay 10, 2014 at 9:06
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$\begingroup$ See also: How to find $\sum_{n=0}^{\infty}\frac{1}{2^{2^n}}$?, Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$ $\endgroup$– Martin SleziakAug 25, 2017 at 12:26
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1 Answer
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By one of the Liouville theorems, this number is transcendental, other similar constructs are $\sum 10^{-n^2}$ and $\sum 10^{-n!}$, or in this context, $\sum 2^{-n^2}$ and $\sum 2^{-n!}$.
So there is no nice formula for this series.
answered May 10, 2014 at 10:30
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$\begingroup$ $\sum_{n=0}^\infty \frac1{n!}$ is transcendental, but it has a nice closed form in my opinion. $\endgroup$ Aug 25, 2017 at 20:16
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$\begingroup$ @SimplyBeautifulArt : No, this number has a commonly used symbol and many applications, but no formula that does not use limits. And no, $\exp(1)$ is also just a name for the exponential power series. $\endgroup$ Aug 25, 2017 at 22:07
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$\begingroup$ Then define "formula", because it seems far too vague and also seems to go against my concept of formula. $\endgroup$ Aug 25, 2017 at 22:09
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$\begingroup$ In this context it is just the confirmation that it can not be expressed by algebraic means. As the name says, these numbers transcend the possibilities of constructive, finite, "nice" formulas. $\endgroup$ Aug 25, 2017 at 22:14
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$\begingroup$ You still haven't answered my question: What is a "formula" here? $\endgroup$ Aug 25, 2017 at 22:16