Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2009 / B / 4 / d / Solution

Choose the blocksIf $p^k\le n<p^{k+1}$, then $|x_n{|}_p=p^{-k}$, so $x_n\to 0$ in ${\mathbb{Q}}_p$. By the ultrametric inequality, for $N\ge M$,The partial sums are Cauchy sequence, and completeness of ${\mathbb{Q}}_p$ proves convergence of $\sum x_n$.

In the real series of p-adic absolute values, the block $p^k\le n<p^{k+1}$ contains $(p-1)p^k$ equal terms of size $p^{-k}$. Its sum is $p-1$, independently of $k$. Infinitely many such positive block sums force divergence. This exhibits p-adic convergence without summable absolute values: the criterion is that the terms tend to zero, rather than real absolute summability.