The real field with convergent generalized power series

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on $[0,1]$ by a series $\sum c_{n} x^{\alpha _{n}}$ with $0 \leq \alpha _{n} \rightarrow \infty$ and $\sum |c_{n}| r^{\alpha _{n}} < \infty$ for some $r>1$ is definable. This expansion is polynomially bounded.