  We study the coefficients of the Taylor series expansion of powers of the
function $\psi(x)=\frac{1-\sqrt{1-x}}{x}$, where the Brunel operator $A\equiv
A(T)$ is defined as $\psi(T)$ for any mean-bounded $T$. We prove several new
precise estimates regarding the Taylor coefficients of $\psi^n$ for
$n\in\mathbb{N}$. We apply these estimates to give an elementary proof that for
any mean-bounded, not necessarily positive operator $T$ on a Banach space $X$,
the Brunel operator $A(T):X\to X$ is power-bounded and satisfies
$\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$ (equivalently, $A(T)$ is a
Ritt operator). Along the way we provide specific details of results announced
by A. Brunel and R. Emilion in \cite{Brunel}.
