Method of finding a p-adic expansion to a rational number Could someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$ 

In the case of $-1/6$, it’s very easy:
$$-\frac{1}{6} = \frac{1}{1-7} = \sum_{k=0}^∞ 7^k.$$
What happens: In the $p$-adic numbers, the sequence $p^k$ is a null sequence (as $|p^k|_p = p^{-k} \overset{k → ∞}\longrightarrow 0$). Because of that and since the $p$-adic distance is ultra metric, this already ensures that the series $\sum_{k=0}^∞ p^k$ is converging – but that isn’t even needed: It’s a geometric series and so
$$\sum_{k=0}^∞ p^k = \lim_{n→∞} \sum_{k=0}^n p^k = \lim_{n→∞}\frac{1-p^{n+1}}{1-p} = \frac{1}{1-p}.$$
This is done for $p = 7$ above.
