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I want to know wht's equal this number :${\frac 12 ^ \frac 13} ^{\cdots \frac1n}$, however it's seems trivial that is converge to 1 , but i don't know how i can evaluate it , Really i have two question: the first is : what is the partial sum of it ? and what's equal for n large enough or go to infty ?

zeraoulia rafik
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    Clarify the order of the brackets. In particular, is the third term $\left(\frac 12\right)^{\left(\frac 13^{\frac 14}\right)}$ or $\left(\frac 12 ^ \frac 13\right) ^ \frac 14$ (which can be simplified, of course)? – Sarvesh Ravichandran Iyer Jun 15 '18 at 04:51
  • (1/2)^(1/3)^(1/4)................^(1/n) – zeraoulia rafik Jun 15 '18 at 04:54
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    I think the convention is that you always do the "higher" exponents first, because (a^b)^c can be written more easily as a^(bc), so that's why a^b^c usually means a^(b^c). – CJD Jun 15 '18 at 04:55
  • I tried playing with some of these terms in Mathematica and to me it doesn't look like it converges to 1. I tried n = 30 and if I did the computation right, the output was .658285. Here is the code I used: n = 30; f[x_] := (n = n - 1; (1/n)^x) NestList[f, 1./n, n - 2] – CJD Jun 15 '18 at 05:04
  • what "partial sum" is there?? – mathworker21 Jun 15 '18 at 05:08
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    @DanielV The sequence $\frac{1}{2}-\frac{1}{n}$ is monotonically increasing in the range $(0,1)$ and has limit $\frac{1}{2}$. – mathworker21 Jun 15 '18 at 05:16
  • How do you know it's monotonically increasing? – CJD Jun 15 '18 at 05:17
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    Also look at: https://math.stackexchange.com/q/1458506/279515 and https://math.stackexchange.com/q/1683961/279515 –  Jun 15 '18 at 05:22
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    @mathworker21 It is definitely not an increasing sequence. It goes up, down, up, down,... The first terms are ${0.5, 0.79\ldots,0.59\ldots,0.73\ldots,\ldots}$. This is easy enough to prove, but the demonstration here in the comments is quicker with the decimals. Whatever the limit is, it is between $0.59\ldots$ and $0.73\ldots$. – 2'5 9'2 Jun 15 '18 at 05:26
  • @alex.jordan There seem to be two accumulation points of around $0.6583656$ and $0.6903471$ – Henry Jun 15 '18 at 08:57

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