Why does 1 + 2 + 4 + 8 + ... converge to -1 in the 2-adic number system?

Question

From this page, http://mathworld.wolfram.com/p-adicNumber.html, the norm of $x = \frac{p^a r}{s}$ for r, s relatively prime to p, and a maximized, over the p-adic numbers, is $p^{-a}$. If we take the norms of $\sum_{n=0}^{\infty} 2^n$ over the 2-adic number system, they are always positive. So how does this lead to the conclusion that the final sum is -1?

Answer 1

The $p$-adic world is totally different from real world. For example a disk (a circle and its inside) has no fixed center. Every point of the disk is a center! Also a circle in $p$-adic world contains balls of same radius in its circumference!! Any triangle is isosceles and is you have a square in $p$-adic world, then the diameter of square is less than its side, and ...

Specially there is no concept of positive or negative $p$-adic number. So it can make sense that a sum of powers of $2$ converges to $-1$.

You also may note that $$\sum_{n=1}^\infty nn!=\sum_{n=1}^\infty(n+1)!-n!\\ =-1+\lim_{n\to\infty} (n+1)!=-1+0=-1$$

$p$-adic objects realm on strange tides.