Timeline for What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$? [closed]
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 4, 2016 at 21:43 | history | closed |
Ian Morris Wolfgang YCor Jan-Christoph Schlage-Puchta Ryan Budney |
Needs details or clarity | |
| Dec 31, 2015 at 15:25 | comment | added | Ian Morris | @FedorPetrov: I see what you mean, perhaps I am mistaken. | |
| Dec 31, 2015 at 12:00 | comment | added | Fedor Petrov | @IanMorris what is so special about $1/n$? It is so in my wrong interpretation, see my answer below and comments, but how is it in the correct setting? | |
| Dec 31, 2015 at 11:06 | answer | added | juan | timeline score: 7 | |
| Dec 31, 2015 at 10:03 | review | Close votes | |||
| Jan 4, 2016 at 21:43 | |||||
| Dec 30, 2015 at 21:08 | comment | added | Ian Morris | The sequence of values $x^{2x^{3x^{4x^{\cdots nx}}}}$ certainly converges for infinitely many values of $x$, since if $x=1/n$ then all terms in this sequence from the $n^{\mathrm{th}}$ onwards are the same. However it is not clear to me that the set of $x>0$ for which this sequence converges has nonempty interior. | |
| Dec 30, 2015 at 21:00 | comment | added | Ian Morris | The convergence of sequences of the form $a_1^{a_2^{a_3^{\cdots a_n}}}$ was investigated by D.F. Barrow in the article ``Infinite Exponentials'' (American Mathematical Monthly 43, No.3 (1936) p.150-160). Perhaps it is intended that $f(x)$ is to be understood as the limit of a sequence of this type. Barrow's results are sufficient to prove the divergence of the sequence of values $x^{2x^{3x^{\cdots nx}}}$ for $x>e^{1/e}$, but do not seem to provide much clear information in other cases. | |
| Dec 30, 2015 at 17:43 | comment | added | Qfwfq | @Ian Morris: maybe it's a transseries arxiv.org/abs/0801.4877 (though, not with "finite exponential height")? | |
| Dec 30, 2015 at 15:53 | comment | added | Ian Morris | I don't wish to seem pedantic, but it isn't obvious to me at a glance what the meaning of this expression is. Perhaps it would be easier to answer the question if you defined $f(x)$ explicitly as the limit of an explicit sequence of functions. | |
| Dec 30, 2015 at 11:09 | answer | added | Fedor Petrov | timeline score: 6 | |
| Dec 30, 2015 at 10:06 | review | First posts | |||
| Dec 30, 2015 at 10:12 | |||||
| Dec 30, 2015 at 10:04 | history | asked | Panglossian Oporopolist | CC BY-SA 3.0 |