So I've seen this technique used before, but can't find where I saw it nor do I remember how to finish. The general idea was let's say you have a recurrence of the form $$a_n=a_{n-1}+\frac{f(n)}{g(a_{n-1})}$$ Where $g$ is an increasing and positive function and $f$ is positive (it's usually something simple like the identity or constant). You want to compute an asymptotic for $a_n$. The idea is that you can rewrite it as $$g(a_{n-1})(a_n-a_{n-1})=f(n)$$ And then sum from $n=1$ to $N$ to get $$\sum_{n=1}^N g(a_{n-1})(a_n-a_{n-1})=\sum_{n=1}^N f(n)$$ Now the LHS looks like a riemann sum, and since $g$ is increasing, we have that it is bounded above by $\int_{a_0}^{a_N} g(x)\, dx$. If $\frac{d}{dx} G(x)=g(x)$, we can say that $$G(a_N)-G(a_0)\geq \sum_{n=1}^N f(n)$$ $$G(a_N)\geq G(a_0)+\sum_{n=1}^N f(n)$$ $$a_N\geq G^{-1}\left(G(a_0)+\sum_{n=1}^N f(n)\right)$$ And this gives a lower bound. But to make any conclusions about the asymptotics we would also need an upper bound and I am stuck there. We can't really use the same approach for the upperbound because we can't make it a right riemann sum.
I'm honestly just curious if anyone has a source for this type of approach or solutions that solve this type of problem this way as I can't remember where I initially saw this approach. But also if anyone knows how to solve it, that would of course also be appreciated
