A question on the 2006 Putnam exam asks for the value of the limit $$\lim_{n\to\infty} \frac{a_{n}^{k+1}}{n^{k}},$$ where $k \ge 2$ is a fixed integer, $a_{0}$ is a fixed positive real number, and $a_{n+1} = a_{n} + a_{n}^{-1/k}$ for $n > 0$.
(SPOILER ALERT)
The solutions posted at Kiran Kedlaya's Putnam Archive mention a heuristic argument due to Richard Stanley for analyzing the asymptotics of $a_{n}$: Simply convert the difference equation $$a_{n+1} - a_{n} = a_{n}^{-1/k}$$ to the differential equation $$y' = y^{-1/k},$$ and look at the asymptotics of $y$. This works beautifully.
My question is: What is the rigorous version of this heuristic argument? Is there a standard theorem or technique being used here? If so, what are the precise conditions required to make the argument work?