[suremarc]

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

[solresol]

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.

[kevinventullo]

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/