Lets say i have the following infinite sum: $$ \sum_n x^n $$ then for $x<1$ we can rewrite this as $$ \frac{1}{1-x} $$ However often I've seen the two as equal in general (granted these are in physics papers) $$ \sum x^n=\frac{1}{1-x} \quad \forall x\in \mathbb{C} $$ with the phrases "as analytic continuation makes this unique" or "as a formal series".
Why is saying 'the analytic ontinuation of a is b' the same as 'a=b'?