I was recently introduced to the $p$-adic numbers, and have been asked to show that for $n > 0$, the $p$-adic expansion of $\frac{1}{1 - p^n}$ is $\sum_{i=0}^\infty p^{in}$.
Could someone tell me whether my solution is correct? It seems right to me, but I don't have much intuition for how these objects work yet:
Observe that for all $N$ we have $\sum_{i=0}^N p^{in} = \frac{1-(p^n)^{N+1}}{1-p^n}$. Taking the limit in the $p$-adic topology as $N \to \infty$, we obtain $\sum_{i=0}^\infty p^{in} = \frac{1}{1-p^n}$ as desired.