### Abstract:

n this thesis the analytic properties of power series over a class of non-Archimedean field extensions of the real numbers, a representative of which will be denoted by F, are investigated. In Chapter 1 we motivate the interest in said fields by recalling work done by K. Shamseddine and M. Berz . We first review some properties of well-ordered subsets of the rational numbers which are used in the construction of such a field F. Then, we define operations + and * which make F a field. Then we define an order under which F is non-Archimedean with infinitely small and infinitely large elements. We embed the real numbers as a subfield; and the embedding is compatible with the order. Then, in Chapter 2, we define an ultrametric on F which induces the same topology as the order on the field. This topology will allow us to define continuity and differentiability of functions on F which we shall show are insufficient conditions to ensure intermediate values, extreme values, et cetera. We shall study convergence of sequences and series and then study the analytical properties of power series, showing they have the same smoothness properties as real power series; in particular they satisfy the intermediate value theorem, the extreme value theorem and the mean value theorem on any closed interval within their domain of convergence.