That is what happens when you play with divergent series… Fun stuff but not easy to explain (if at all).

The infinite sum involving powers of two is actually true in the 2-adic integers.

And any of his other examples would work with powers-of-p if p is prime.

Those infinite series aren't divergent if you are measuring their size p-adically.

Gouvea's book actually goes through the things that RobAlni has just discovered.

Yep, and that last number is the 2-adic representation of -4/3.

It turns out a lot of ideas from calculus and real analysis carry over to the 2-adics, and this can lead to some interesting computational shortcuts since two’s complement is essentially native 2-adic arithmetic. I’ve blogged a bit about this here: https://kevinventullo.com/

check out https://zjkmxy.github.io/posts/2021/11/twos-complement-2-adi... for a discussion of some relevant mathematics; cool stuff

there is also a lot of good stuff concerning such computational tricks and their relation with 2-adics in section 7.1.3 (Bitwise Tricks and Techniques) of volume 4A of The Art of Computer Programming: https://www-cs-faculty.stanford.edu/~knuth/fasc1a.ps.gz