The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the <A HREF="http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf">Emden–Fowler nonlinear differential equation</A>,
$$f''(t)=t^{p}[f(t)]^q,$$
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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The book mentioned in a comment, <A HREF="https://link.springer.com/chapter/10.1007/978-3-030-77139-3_1">Sequences of Real Numbers</A>, by Sîntămărian & Furdui, presents on page 11 the recursion $$x_{n+1}=x_n+\frac{n^{2\beta}}{x_1+x_2+\cdots+x_n}$$ and the derivation of the limit $$\lim_{n\rightarrow\infty}x_n/n^\beta=\sqrt{1+1/\beta}$$ as an application of the <A HREF="https://en.wikipedia.org/wiki/Stolz–Cesàro_theorem">Stolz–Cesàro Theorem.</A>