On convergence and rational summation of power series in p-adic field

TL;DR

This paper explores the convergence and rational summation of power series in p-adic and real norms, introduces summation formulas, and investigates their applications in adeles and ideles.

Contribution

It provides new insights into the convergence criteria and rational summability of power series in p-adic and real contexts, including applications to adeles and ideles.

Findings

Established convergence conditions for power series in p-adic and real norms

Derived summation formulas for special power series

Connected power series convergence to adelic and idelic structures

Abstract

In this paper we have discussed convergence of power series both in p-adic norm as well as real norm. We have investigated rational summability of power series with respect to both p-adic norm and real norm under certain conditions. Then we have studied convergence of specially constructed power series and derived summation formula. Finally, we have studied the adele, idele and some results regarding it with the help of convergent power series.