The sum or product of two power series that converge on $\abs{x}<R$
also converges on $\abs{x}<R$, due to absolute convergence, say.
(and maybe bigger (see below)).
What if they have different/overlapping discs of convergence?
Then they clearly converge on the overlap, but they may converge on a
bigger set: to figure it out, you should look at the analytic
functions, and zeros and poles.

For instance, a sum of power series will converge on a bigger set if
poles cancel; a product will converge on a bigger set if zeros and
poles cancel.

Notice how this -forces- you to look at analytic functions over $\bC$:
otherwise the behaviour looks random (if you just try to understand
them in terms of the sequences of coefficients).

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power series over ring: invertible iff constant term a unit

power series are the local ring par excellence

power series in $(x-a)$ aren't silly: they're the local ring at $a$
