  New bounds for the $k$-th order derivatives of the solutions of the normal
and multivariate normal Stein equations are obtained. Our general order bounds
involve fewer derivatives of the test function than those in the existing
literature. We apply these bounds and local approach couplings to obtain an
order $n^{-(p-1)/2}$ bound, for smooth test functions, for the distance between
the distribution of a standardised sum of independent and identically
distributed random variables and the standard normal distribution when the
first $p$ moments of these distributions agree. We also obtain a bound on the
convergence rate of a sequence of distributions to the normal distribution when
the moment sequence converges to normal moments.
