Motivation: In Spanier's Algebraic Topology we are introduced to direct and indirect limits in the first few pages. I partially understand the idea behind it but since I am an example-driven learner I'd actually like to compute some direct limits.
Question: Where can I find explicit examples of a direct (or even indirect) limit being calculated?
What I've done so far: So far, I've computed the direct limit for some finite directed sets. On the wikipedia page, there are a number of examples which I've been trying to work out. In particular, the third example (which is the infinite chain of ${\mathbb Z}/p^{n}{\mathbb Z}\rightarrow {\mathbb Z}/p^{n+1}{\mathbb Z}$ with multiplication by $p$ is the map) I'm unsure of how they come up with the roots of unity for some power of $p$. Any help on this one would also be greatly appreciated.