We think about a relation between a function $\psi(x,\epsilon)$ and a divergent power series in $\epsilon$: $f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{p}+...$ The coefficients $f_{0}$, $f_{1}$... can be a function of $x$ only or of $x$ and $\epsilon$.
Now, let: $\psi_{p}=f_{0}+\epsilon\,f_{1}+...+\epsilon^{p}f_{p}+...$
If $\lim \limits_{\epsilon \to 0}(\psi-\psi_{p})\epsilon^{-p}=0$
Then, the series can be an asymptotic representation of the function $\psi$ and I can say $\psi(x,\epsilon)=f_{0}+\epsilon\,f_{1}...$
How do I get the idea?
- Why would it be beneficial to think of the coefficients as functions of $x$ only? If $\psi_{p}$ is a function of $x$ and $\epsilon$, why would the coefficients be a function of $x$ only?
- Is $\psi_{p}$ introduced in order to compare $\psi_{p}$ and $\psi$?
- Why does $\epsilon^{-p}$ appear in the asymptotic equality?
I am trying to develop an understanding. Please let me know if I am being unclear or not specific enough. Is there maybe another concept I should understand before returning to my questions?