Convergence of P-adic series How can I prove that $\sum_{i = 0}^\infty 2^i$ doesn't converge with respect to $|\cdot|_3$ without resorting to the fact that $\sum a_i$ converges wrt $|\cdot|_p$ if and only if $|a_i|_p \to 0$? Is it even possible?

The only thing that $\sum_0^\infty 2^n$ can converge to is $-1$. But
this series does not converge $3$-adically to $-1$. If it did then
$$\left|-1-\sum_{n=0}^N 2^n\right|_3<1$$
for all large enough $N$. This means that
$$\sum_{n=0}^N 2^n\equiv-1\pmod 3\tag{*}$$
for all large enough $N$. But $3\nmid 2^n$ for all $n$, so if (*) holds
for $N$, it fails for $N+1$. Contradiction.
