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> 

The OP asks where this recursion relations 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>,
$$f''(t)=t^{p}[f(t)]^q,$$
with $p=2\beta$, $q=-1$.