I am trying to understand (and then explain) analytic continuation as simply as possible.
Question 1: Is the following an example of analytic continuation?
The following function is defined for all $x$ except $x=1$. The following illustrates this function.
$$f(x)=1/(1-x)$$
The following series is a representation of $f(x)$ but only valid for $-1<x<1$. It is in fact the Taylor series for $f(x)$ developed around $x=0$.
$$ S_0(x) = 1+x+x^2+x^3+x^4+\ldots$$
The following series is another representation of $f(x)$ but this time valid for $1<x<5$. It is in fact the Taylor series for $f(x)$ developed around $x=3$.
$$S_3(x)=-\frac{1}{2}+\frac{1}{4}\left(x-3\right)-\frac{1}{8}\left(x-3\right)^2+\frac{1}{16}\left(x-3\right)^3-\frac{1}{32}\left(x-3\right)^4+\ldots$$
Question 2: Can we say that $S_3$ is an analytic continuation of $S_0$, where both represent $f(x)$ but in different domains?
Question 3: Does analytic continuation require an overlap of the domains? For the above example, this would require, for example, $S_3$ and $S_5$, the Taylor series developed around $x=3$ and $x=5$.
The following diagram visualises these functions.
