If I am iterating some function $f$ that goes to infinity as x goes to infinity with error $o(g(x))$, for example, is there anyway to bound the error? To be more specific, if I have some sequence $a_n, a_1=1, a_{n+1}=f(a_n)+o(g(a_n))$, is there something we can say about how well $f^{k-n}(a_n)$? approximates $a_k$?
$f^{k-n}$ is f itterated $k-n$ times.
More clarification: If a sequence $a_1, a_2, a_3,...$ is modeled by $f(1), f(f(1)), f(f(f(1))),...$, what can say about the behavior of the sequence $a_1-f(1), a_2-f(f(1)),...$? Can we say that this sequence is o(some function that depends on g)?
This is supposed to be a comment but I can't fit it into the comment box.
Even if you have $f(x) = h(x)$ for all $x > $ some $M$ and $|f(x)-h(x)| \ll 1$ over $[0,M]$, the two sequences $(x_n)$, $(y_n)$ defined by $$ x_n = f^{\circ (n-1)}(1) = \begin{cases}1, &n = 1\\ f(x_{n-1}), & n > 1\end{cases} \quad\text{ and }\quad y_n = h^{\circ (n-1)}(1) = \begin{cases}1, &n = 1\\ h(y_{n-1}), & n > 1\end{cases} $$ can have very different behavior. For a concrete example, let $\epsilon$ be a small number, say $\epsilon = 0.001$ and define
$$f(x) = x(2-\cos(2\pi x))+1 \quad\text{ and }\quad h(x) = f(x) + \begin{cases} \epsilon x(2-x),& x < 2\\0, &x \ge 2\end{cases} $$ It is easy to check $x_n = f^{\circ (n-1)}(1) = n$ and yet $y_n = h^{\circ(n-1)}(1)$ blows up non-linearly! $$\begin{array}{|r:r|} \hline n & y_n\\ \hline 1 & 1.0000000000\\ 2 & 2.0010000000\\ 3 & 3.0010394980\\ 4 & 4.0011035079\\ 5 & 5.0011996822\\ 10 & 10.0035221944\\ 15 & 17.5415709482\\ 20 & 463.3074993684\\ 25 & 32972.6418308000\\ 30 & 2651994.7780103278\\ 35 & 15400024.3406490520\\ 40 & 857042166.1092528100\\ 45 & 44608271060.7540660000\\ 50 & 713883229051.1130400000\\ \hline \end{array}$$