Rate of convergence of roots

Question

Suppose I have a sequence of smooth functions $f_n$ with unique roots $x_n$, such that $f_n$ converges pointwise to a smooth function $f$ with unique root $x$. Also assume that $x_n\to x$. Is there a way to describe the rate of convergence of $x_n$ to $x$, possibly in terms of derivatives of $f_n$ and $f$?

Answer 1

Instead of keeping things general, start with a simple example.

Let the sequence of smooth functions $(f_n)_{n\in \mathbb{N}}$ be defined by $f_n(x) = (x-\frac{1}{n})(x + \frac{1}{n})(x-\frac{2}{n})(x+\frac{2}{n})$. This sequence has a limit $f(x) = x^4$, which has a unique root $x^{\ast}=0$. This adheres to the parameters of your question ($f_n$ has multiple roots and all of them converge to a single root of $f$ as $f_n$ converges to $f$).

The sequence $f_n$ has $4$ roots $ \frac{1}{n},-\frac{1}{n},\frac{2}{n},-\frac{2}{n}$. Taking derivatives of $f_n$ gives $f_n^{\prime}(x) = 4x^3 - \frac{10}{n^2}x$, $f_n^{\prime \prime}(x) = 12x^2 - \frac{10}{n^2}$, $f_n^{\prime \prime \prime}(x) = 24x$.

In this example, can you explain the rate at which the roots converge depending on the rate at which the derivatives of the sequences converge?