On convergence of power series in p-adic field

In this paper we have discussed about the region of convergence of power series in p-adic field. we have investigated some sufficient conditions for which a power series has same radius of convergence with respect to both p-adic absolute value and usual absolute value on the field of rationals $\mathbb{Q}$ i.e., in p-adic field and real field. Next, we have characterized the rationals at which the power series converges both in real norm and p-adic norm within the same radius of convergence. More importantly, we have investigated some power series having rational sums both in real norm as well as p-adic norm. At the end we have studied the Adelic aspect and some results regrading it.