The model theory of tame fields in the language of valued fields has been extensively studied by Kuhlman. In particular, the theory of an equal characteristic tame field in this language is given by the theory of the residue field and the theory of the value group. Building on Kuhlman's results, we give an AKE-principle for tame valued fields of equal characteristic in L_t, the language of valued fields with a distinguished constant symbol t. Furthermore, we use this principle together with Kedlaya's work on the connection between generalised power series and finite automata to show that a tame Hahn field of equal characteristic is decidable in L_t if it has decidable residue field and decidable value group. In particular, we obtain decidability of F_p((t^{1/p^\infty})) and F_p((t^Q )) in L_t. Time permitting, we will also see how approximation methods used in this work reveal a condition on algebraicity for generalised power series in terms of the order type of the support.