  The renormalized trajectory of massless $\phi^4$-theory on four dimensional
Euclidean space-time is investigated as a renormalization group invariant curve
in the center manifold of the trivial fixed point, tangent to the
$\phi^4$-interaction. We use an exact functional differential equation for its
dependence on the running $\phi^4$-coupling. It is solved by means of
perturbation theory. The expansion is proved to be finite to all orders. The
proof includes a large momentum bound on amputated connected momentum space
Green's functions.
