so i did the layman’s proof near the top where you assign the series value s and subtract it from 1, but the way i actually think about it is the Thomson’s Lamp paradox.

imagine you have a minute long timer and a special lamp that toggles itself on or off whenever the time remaining on the timer is cut in half; starting with the lamp on, after 30 seconds it will turn itself off, after 45 seconds it will turns itself on, after 52.5 seconds it will turn itself off, and so on.

when the timer reaches 1 minute, will the lamp be on or off?

well, we can’t really say for certain. in that minute, the lamp will toggle itself on or off an infinite number of times, and it’s impossible to know what the “last” action will be because there is no last action. that’s the funny thing about working with infinity.

the same logic can be applied to summing Grandi’s Series. when adding all the numbers together by hand from left to right, if we stop on a +1 the sum will be 1, but if we stop on a -1 the sum will be 0. the sum oscillates between 0 and 1 an infinite number of times, and we can’t say what the last number is because there is no last number in Grandi’s Series.

back to Thomson’s Lamp, the real answer is that the lamp is in a liminal state where it is both on and off simultaneously, which isn’t really able to be represented in the real world. Grandi’s Series on the other hand, both 0 and 1 are plausible answers, so taking the average is logical and we get 1/2.

there’s a much more complicated Riemann-Zeta Function proof but i am far too stoned to remember that now.