Relationship between analytic continuity and 2-adic convergence of the sum of powers of 2

Question

Using the 2-adic metric for natural numbers, we have $$ \sum_{n=0}^\infty 2^n =^2 -1 $$ where $=^2$ is used to denote convergence of the partial sum with respect to the 2-adic metric. On the other hand, in the usual Euclidian metric we have $$ \sum_{n=0}^\infty p^n = \frac{1}{1-p}$$ for complex numbers $\left| p \right| < 1$. The right-hand function has a unique analytic continuation to all of $\mathbb{C}\setminus\{1\}$. Evaluated at $p=2$, this gives the 'relationship' $$ \sum_{n=0}^\infty 2^n =^{a} -1 $$ where $=^a$ is used to denote the evaluation provided through analytic continuity. What is the reason for this connection between 2-adic convergence of the sum and its evaluation through analytic continuity? Why do they both give $-1$?