I apologize in advance if this is a very stupid question.
Analogy: When people were working on solving cubics, it came out in a certain class of cubic equations that even if the final answers were all real numbers, intermediate calculations still required complex numbers. In a sense you had to pass through some other field in order to get from your own field back to your own field.
Are there similar examples in power series? For example, we start with some expression with convergent power series, use some manipulations that are only really justified for formal power series as there is no radius of convergence, but then end up back with a convergent power series?