  We present a convergence proof for higher order implementations of the
projective integration method (PI) for a class of deterministic multi-scale
systems in which fast variables quickly settle on a slow manifold. The error is
shown to contain contributions associated with the length of the microsolver,
the numerical accuracy of the macrosolver and the distance from the slow
manifold caused by the combined effect of micro- and macrosolvers,
respectively. We also provide stability conditions for the PI methods under
which the fast variables will not diverge from the slow manifold. We
corroborate our results by numerical simulations.

Numerical Analysis