In real analysis, I am learning about convergence of series in metric spaces. My professor states that in the metric space of $\mathbb Z$ with the $2$-adic metric, one of the series converges and the other does not:
$$ \sum_{n=0}^\infty 2^n$$ $$ \sum_{n=0}^\infty (-2)^n$$
I know that the first series converges to $-1$, so his claim is that the latter series does not converge. However, this series is Cauchy and $\lim_n (-2)^n=0$, and I am under the impression that this implies the series converges in the $2$-adics.
Both of these series definitely converge in $\mathbb{Z}_2$. As you say, the sequences of partial sums are both Cauchy.
However, the second series converges to an element of $\mathbb{Z}_2$ which is not an element of $\mathbb{Z}$; namely, $1/3$. To see this, use the old geometric series trick: $$ \frac{1}{3} = \frac{1}{1 - (-2)} = 1 + (-2) + (-2)^2 + \ldots $$