Suppose that we have some sequence $(a_n)_{n\ge 1}$ and its generating function $\displaystyle f(x)=\sum_{n=1}^{\infty} a_{n}x^n$. How can I find the generating function of $(a_{2n})_{n\ge 1}$ from this?
I know how to do this if we consider the generating function to be a formal power series. But I don't want to think of it as being formal. I want to think of it as a "proper" power series.
So, I am regarding my function $f$ as being defined on some interval $(-r, r)$ (where $r>0$ is the radius of convergence of that power series). For $x$ in this range I have $$\sum_{n=1}^{\infty} a_{2n}x^{2n}=\frac{1}{2}(f(x)+f(-x)).$$
For $x\in [0, r)$, I can just change $x$ to $\sqrt{x}$ in the relation from above, but what do I do for $x\in (-r, 0)$? I know that if I work with formal power series I have no problem whatsoever, but I want to do this with "proper" power series. Alternatively, I would like to know how to pass from formal power series to "proper" ones (i.e., if I do this with formal power series, how do I know for which $x$ it works?).