What does it mean for a power series to be 'defined' in this context?

Question

The following screenshot is taken from some lecture notes on combinatorics and discusses the possible ways of expressing a power series

where $(5.2)$ refers to the power series given by the expansion of $$ (1+x+x^2+\dots)(1+x^2+x^4+\dots) $$

As stated in the screenshot, $f(x)$ and $g(x)$ will converge iff $|x| < 1$. Does this mean that the expansion $(5.2)$ is only defined for $|x| < 1$?

Slightly further on in the lecture notes, the following is said:

However, Wikipedia says that

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or $\infty$.

So what does it mean in my lecture notes when it say "If [the power series] has a positive radius of convergence..."? Surely this is a given, based on the Wikipedia definition?

Answer 1

Given two formal series with coefficients in a conmutative ring $\mathbb{A}$ (such as $\mathbb{C}$),

$$ f = \sum_{n\geq0}a_nX^n \ , \ g = \sum_{n\geq0}b_nX^n \in \mathbb{A}[[X]] $$

you can define their product as the Cauchy product,

$$ fg = \sum_{n\geq0}(\sum_{k=0}^na_kb_{n-k})X^n $$

Now, if $\mathbb{A} = \mathbb{C}$ and we see each series as the complex function they define, $f$ and $g$ converging over some set does not imply that the product converges, and therefore (as a function) $fg$ will be defined only if the series of the Cauchy product of $f$ and $g$ is indeed convergent on some disk, and that is the clarification that is being made in the text.

Answer 2

If $f$ and $g$ are analytic in domains $\Omega_f$, $\Omega_g$ containing the origin then the function $$h(z):=f(z)\>g(z)$$ is analytic at least in the intersection $\Omega_f\cap\Omega_g$. If furthermore $$f(z)=\sum_{j=0}^\infty a_j z^j\quad\bigl(|z|<\rho_f\bigr),\qquad g(z)=\sum_{k=0}^\infty b_k z^k\quad\bigl(|z|<\rho_g\bigr)$$ then $$h(z)=\sum_{r=0}^\infty c_r z^r\qquad\bigl(|z|<\min\{\rho_f,\rho_g\}\bigr)\ ,$$ whereby $$c_r:=\sum_{k=0}^r a_{r-k}b_k\ .$$