Two's complement unexpectedly shows up in infinite series

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

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

check out <a href="https:&#x2F;&#x2F;zjkmxy.github.io&#x2F;posts&#x2F;2021&#x2F;11&#x2F;twos-complement-2-adic&#x2F;" rel="nofollow">https:&#x2F;&#x2F;zjkmxy.github.io&#x2F;posts&#x2F;2021&#x2F;11&#x2F;twos-complement-2-adi...</a> for a discussion of some relevant mathematics; cool stuff<p>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: <a href="https:&#x2F;&#x2F;www-cs-faculty.stanford.edu&#x2F;~knuth&#x2F;fasc1a.ps.gz" rel="nofollow">https:&#x2F;&#x2F;www-cs-faculty.stanford.edu&#x2F;~knuth&#x2F;fasc1a.ps.gz</a>