Analysis

Suppose that the function $f$ has a zero at $a$, i.e., $f(a) = 0$.

If $f$ is continuously differentiable and its derivative does not vanish at $a$, then there exists a neighborhood of $a$ such that for all starting values $x_0$ in that neighborhood, the sequence $\{x_n\}$ will converge to $a$.

In practice this result is ``local'' and the neighborhood of convergence is not known a priori, but there are also some results on ``global convergence.'' For instance, given a right neighborhood $U$ of $a$, if $f$ is twice differentiable in $U$ and if $f' \ne 0$, $f \cdot f'' > 0$ in $U$, then, for each $x_0\in U$ the sequence $x_k$ is monotonically decreasing to $a$.