One of the problems Mark wanted us to be ready for was to outline the proof of the following:

If $\sum a_n x^n$ converges absolutely at $x_0$, then it converges uniformly on $[-c,c]$, whenever $ 0\leq c<|x_0|$.

I will go ahead and write out my outline; since we're calling it an outline, I'd love not only thoughts on logic, but also how much detail I'm including.

By assumption, we know that $\sum a_n x_0^n$ converges absolutely. Since $x \in [-c,c]$ are strictly less in absolute value than $x_0$, we know that $|a_n x^n|<|a_n x_0^n|$, and so we can apply the Weierstrauss $M$ test with $M_n = a_0 x_0^n$. Therefore, $\sum a_n x^n$ converges uniformly on $[-c,c]$.

I know that didn't fully flesh things out, but do you guys think that works as a rough outline?