The theorem I'm referring to is as follows:
Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated at $z_0$ differs from zero. Then the power series of $f$ about $z_0$ begins with the $n$-th term, and $f$ is said to have a root of multiplicity (or “order”) $n$. If $n = 1$, the root is called a simple root.
What is the intuition behind this theorem?