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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2There 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. – achille hui May 10 '14 at 09:06
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See also: How to find $\sum_{n=0}^{\infty}\frac{1}{2^{2^n}}$?, Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$ – Martin Sleziak Aug 25 '17 at 12:26
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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.
Martin Sleziak
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Lutz Lehmann
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$\sum_{n=0}^\infty \frac1{n!}$ is transcendental, but it has a nice closed form in my opinion. – Simply Beautiful Art Aug 25 '17 at 20:16
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@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. – Lutz Lehmann Aug 25 '17 at 22:07
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Then define "formula", because it seems far too vague and also seems to go against my concept of formula. – Simply Beautiful Art Aug 25 '17 at 22:09
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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. – Lutz Lehmann Aug 25 '17 at 22:14
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You still haven't answered my question: What is a "formula" here? – Simply Beautiful Art Aug 25 '17 at 22:16
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Under contexts such as here, it is generally agreeable that a form involving the constants $e$, $\pi$, and elementary functions is considered closed form. Sometimes a broader or more restrictive range of constants and functions may be considered, though you've made no mention of the such other than now a hinting at a restriction to algebraic functions. What in the (very short) question makes you think a restriction to algebraic functions was called for? – Simply Beautiful Art Aug 25 '17 at 22:19
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Of course you can say "I define the value of this series as 'bla'" and the the answer is 'bla', but then this does not give any more information on this series. There are more transcendental numbers than names for constants, and that holds even including ´ algebraic expressions of named constants, so it is quite save to say that any random transcendental number does not have an expression in named transcendental constants. Of course then one could argue that one can name any transcendental ever written down, however we lose sight of the intent of the question. – Lutz Lehmann Aug 26 '17 at 04:57
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But what's to say it doesn't have a closed form involving $e$ and $\pi$? As I said, closed form usually permits these constants. I'm not asking for you to name every transcendental number and use them, I'm only asking for $2$ of them. (And $e$ is not necessarily defined by that series, though it can be) – Simply Beautiful Art Aug 26 '17 at 12:00