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I came accross a curiosity I don't fully understand. In an exam, the sequence $\left(x_n\right)$ is introduced such as for all $n \geq 1$ $$ x_{n+1}=\left(1+\frac{1}{x_n}\right)^n $$ with $x_1 = \alpha \in \left]0;+\infty\right[$. It is argued that there exists one number $\alpha^{\ast}$, and only one, such that the sequence $\left(x_n\right)$ with $x_1 = \alpha^{\ast}$ diverges towards $+\infty$. A proof is then proposed where it is shown that $\alpha^{\ast} \approx 1,1874$. This result triggers my curiosity.

I've written a small python script, and I've observed that as soon as $x_{N}$ is "big", then $x_{N+1} \approx 1$. I guess all the mystery is there : how to have $x_{N}$ 'big' without having $x_{N+1} \approx 1$. For $x_{N}$ big enough, we can write

$$ x_{N+1} = e^{N\ln\left(1+\frac{1}{x_N}\right)} = e^{N\left(\frac{1}{x_{N}} - \frac{1}{2x_{N}^2} + o\left(\frac{1}{x_{N}^2}\right)\right)} $$ So my guess is that $N/x_{N}$ needs to stay low. What I see for example is :

\begin{array} $x_1 & 5 & 1.1874\newline x_2 & 1.20 & 1.84\newline x_3 & 3.36 & 2.38\newline x_4 & 2.18 & 2.86\newline x_5 & 4.52 & 3.31\newline x_6 & 2.72 & 3.74\newline x_7 & 6.55 & 4.14\newline x_8 & 2.70 & 4.54\newline x_9 & 12.4 & 4.92\newline x_{10} & 2.0 & 5.29\newline \end{array}

A pattern clearly emerges with $\alpha = 1.1874$, where odd and even indexes find a kind of balance. But after some more iterations, the balance is broken : we find $x_{28} = 3.2 \cdot 10^{8}$ and $x_{29} \approx 1.0$. Does it mean that it is not possible for a program to reproduce this pattern and just preserve it until a given rank ? Is there an equation that could be deduced for $\alpha^{\ast}$ ?

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  • $\begingroup$ Do you have a link to this proof and how they compute $a^*$? I'd imagine that the result is very unstable and requires a specific irrational number $\endgroup$
    – wjmccann
    Apr 19, 2023 at 19:43
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    $\begingroup$ That is en.wikipedia.org/wiki/Foias_constant $\endgroup$
    – Martin R
    Apr 19, 2023 at 19:43
  • $\begingroup$ @MartinR Thanks ! I did not know that it existed ! $\endgroup$
    – Atmos
    Apr 19, 2023 at 19:50
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    $\begingroup$ Apparently the precision of built-in floating point numbers is not enough for the late-20s terms. Consider using the "decimal" module if you want more precision. $\endgroup$ Apr 20, 2023 at 1:23

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