Convergence of some perturbed sequences of rational powers and application to syracuse problem

Sequences of rational powers \left( \xi\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse, Z-number and waring problems. Conjectures from such problems are known to be intractable and only few partial results exist until now. In this paper, we study a family of perturbed sequences of rational powers called 'Branch sequences' of the form \left( S_{n}=\left( \xi+\Sigma_{n} \right)\left( \frac{p^{n}}{q^{n+e_{n}}} \right) \right)_{n\ge 0}. Under the assumption that such sequences are deterministic and they have controlled positive perturbations, we establish the convergence result: min_{n\ge 0}(S_{n})\le q^{2}. As an application, we show that Syracuse sequences are 'Branch sequences' with all the required conditions for convergence and therefore this confirms the Collatz conjecture. Keywords: Sequences of rational powers, Syracuse conjecture, Collatz problem, 3x+1 problem.