[jampekka]

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

[madcaptenor]

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".

[foldU]

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