For that, I would look at early calculus/pre-calc, I think, examining infinite series and their properties and equivalencies.

There's certain forms like that that have well known values that they converge to as you continue adding terms into infinity. Sometimes that convergence is only possible if your domain is limited, eg. [0,1].

That clears it, thanks! I didn't figure out that "sensible" referred to convergent G(t).

It doesn't mean that in general; I think the author is saying "for sensible v" somewhat informally to mean "for v for which this makes sense".

You also don't really have to worry about convergence, since these are formal power series.

This is just the sum of a geometric sequence.