Convergence on the Levi-civita Field and Study of Power Series

Convergence under various topologies and analytical properties of power series on Levi-Civita fields are studied. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points the distance of which from the center is infinitely smaller than the radius of convergence. In addition to allowing the introduction of common transcendental functions, power series are shown to behave similar to real power series. Besides being infinitely often differentiable and re-expandable around other points, it is shown that power series satisfy a general intermediate value theorem as well as a maximum theorem and a mean value theorem.