Does $\sum n $ converge p-adically?

Question

Does $\sum n $ converge p-adically, I have worked out $v_p(n) \leqslant log(n)/log(p) $

not sure how to conclude from this

I want to prove this using the result that it converges p-adically iff $v_p(n)$ tends to infinity as n tends to infinity

Answer 1

A series $\sum a_n$ converges $p$-adically if and only if $a_n$ tends to zero $p$-adically, because the $p$-adic absolute value is non-archimedean. Now, the sequence $n$ does not tend to zero $p$-adically, as its extract $1+p^n$ does not converge to zero, as it has constant $p$-adic absolute value equal to $1$.

Answer 2

No -- whenever you add something that is not a multiple of $p$, the $p$-adic distance between the previous and current partial sum is $1$.

So the sequence of partial sums can't be Cauchy.