Recently, I encountered this problem:  

*"Given a sequence of positive number $(x\_n)$ such that for all $n$,
$$x\_{n+1}=x\_n+\frac{n}{x\_1+x\_2+\cdots+x\_n}$$
Find the limit $\lim\_{n \rightarrow \infty} \frac{x\_n}{\sqrt{n}}.$"*  

As pointed out in the comment section of the following topic [[1](https://math.stackexchange.com/questions/4488666/find-lim-n-to-infty-fracx-n-sqrtn-where-x-n1-x-n-fracnx-1x)], this problem is indeed open. Out of curiosity, I wonder if this problems has a research-level meaning? For example, does it stem from an interesting system dynamic question?   

  
I'm aware this question might not suitable for Mathoverflow forum, but aside from this site, I don't think there is a more suitable place to ask this question



"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the [Emden–Fowler nonlinear differential equation](http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf),
$$f''(t)=t^{p}[f(t)]^q,$$
for $p=1$, $q=-1$. A particular solution is
$$f(t)=\lambda t^{(p+2)/(1-q)},\;\;\lambda=\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$
One readily checks that the asymptotic limit $x\_n\rightarrow \sqrt{3n}$ for $x\_n=f'(n)$ is obtained for $p=1$, $q=-1$.




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*This connection to the Emden-Fowler equation motivates the following*  

**Conjecture:** The recursion relation
$$x\_{n+1}=x\_n+n^{p}(x\_1+x\_2+\cdots+x\_n)^q$$
has for $p\geq 0$, $q\leq 0$, $p+q>-1$, the limit
$$\lim\_{n\rightarrow\infty}n^{(1-q)/(1+p+q)}x\_n=\frac{p+2}{1-q}\left[\frac{(p+2)(p+q+1)}{(q-1)^2}\right]^{1/(q-1)}.$$


For $q=-1$ this is stated as an open problem on page 11 of [Sequences of Real Numbers](https://link.springer.com/chapter/10.1007/978-3-030-77139-3_1), by Sîntămărian & Furdui.


Here are two numerical tests (blue is the recursion for $x\_n$, gold the conjectured large-$n$ asymptotic):


![](https://i.stack.imgur.com/Is6dl.png)
![](https://i.stack.imgur.com/SPYwm.png)
