Suppose I have an analytic function $f : \mathbb{R} \to \mathbb{R}$ and I have the asymptotic expansion of some $x\_0$ up to a few terms in terms of $\epsilon$ for some $\epsilon \to 0$ which I believe is such that $f(x\_0) = 0$. How can I prove that $$f(x\_0) = 0$$


Some context - In my case, I am trying to compute the asymptotic expansion of the solution to some saddle point equation of form $f(x) = 0$, in order to apply the saddle point method from Asymptotic Analysis/Analytic combinatorics/analytic number theory. However, actually computing the solution to this equation seems difficult. Instead, I want to 'guess' the solution to the equation, and prove that my guess is correct. Is this possible?



How to check if you have the asymptotic solution of some equation?
Let $\bar x\_0$ be your approximation of $x\_0$. Find a small $\delta$ such that $f(\bar x\_0)$ and $f(\bar x\_0+\delta)$ have opposite sign. Then you know $x\_0$ to an accuracy of $|\delta|$ (assuming $f$ is continuous and monotonic as it usually is in the saddle-point method).


More generally, show that Newton's method converges or find a contraction mapping.


