Convergência de séries p-ádicas

Resumo

Accustomed to working with the real numbers and the usual notion of distance, we wonder at the impact that a change in the metric can cause. Given a prime p, the p-adic norm of a rational number measures its size with respect to the powers of p, declaring that an irreducible fraction is small if its numerator is divisible by a high positive power of p. This arithmetic method to estimate distances is quite different from the com- mon absolute value and, in particular, the family of convergent sequences is distinct from the real one. We will concern ourselves with the existence of series with rational terms whose convergence holds no matter the chosen norm and such that, though their limits vary with the metric, we know how to master this dependence.