p-adic convergence without summable absolute values
= p-adic convergence without summable absolute values
{c}
{title2=$x_n=p^{\lfloor\log_p n\rfloor}$}
In a complete <Non-Archimedean absolute value> <field>, a series converges if and only if its terms tend to zero: the <ultrametric inequality> bounds every finite tail by its largest term. The displayed <sequence> tends to zero in $\mathbb Q_p$. But each block $p^k\leq n<p^{k+1}$ contributes $p-1$ to the real sum of its <p-adic absolute values>, so that real sum diverges.