The trivial connection contribution to Witten’s invariant and finite type invariants of rational homology spheres

We derive an analog of the Melvin-Morton bound on the power series expansion of the colored Jones polynomial of algebraically split links and boundary links. This allows us to produce a simple formula for the trivial connection contribution to Witten’s invariant of rational homology spheres. We show that thenth term in the 1/K expansion of the logarithm of this contribution is a finite type invariant of Ohtsuki order 3n and of at most Garoufalidis ordern.