  We establish estimates and representations for the remainders of Taylor
approximations of the spectral action functional $V\mapsto\tau(f(H_0+V))$ on
bounded self-adjoint perturbations, where $H_0$ is a self-adjoint operator with
$\tau$-compact resolvent in a semifinite von Neumann algebra and $f$ belongs to
a broad set of compactly supported functions including $n$-times differentiable
functions with bounded $n$-th derivative. Our results significantly extend
analogous results in \cite{SkAnJOT}, where $f$ was assumed to be compactly
supported and $(n+1)$-times continuously differentiable. If, in addition, the
resolvent of $H_0$ belongs to the noncommutative $L^n$-space, stronger
estimates are derived and extended to noncompactly supported functions with
suitable decay at infinity.
