It all boils down to, I suppose, there's no final step and so the question of whether the lamp is on/off is moot.

@Hudjefa That, in combination with the switch alternating between a 0 and a 1 position. E.g. if you have a "more physical" lamp with the usual inductivity of electrical currents, a 50% current intensity will result; mathematically, the series will become more like an integral over the last switching cycles and converge to a 50% value, while an alternating 0-1 series does not converge. I.e. you need both, an infinite series and nonconvergence.

Now that you brought 0 and 1 to my attention, I recall reading the switch's state is 1/2 or 0.5 ... somewhere between off and on, kinda like quantum superposition, hovering between 2 mutually exclusive states. You can't say it's on and you can't say it's off. Is it neither or is it both? RIP Sanjaya Belathiputta.

The average state of the lamp definitely converges.

The Grandi series is not applicable here because the intervals are deminishing. If they were equal, the finite result would be the intermediate state of 1/2, indeed. In this case, not.

@Anixx The Grandi series here represents to the sequence of states; the intervals do not matter for this. The states On, Off, On, Off, ... can be perfectly represented as +1, -1, +1, -1, ... which is exactly the Grandi series.

Perhaps we should ask the question, does Achilles cross the finish line with his left foot or his right foot? :D

@Anixx The intervals matter only insofar as their diminishing length places the series' "infinity" at a finite point in time, so we need to the series' limit to assign a meaningful state to that time. Turns out the series does not have a limit.

I agree with the interpretation yielded by the Grandi series value of 1/2 being that a superposition of states exists at x=2. I did some numerical simulation in Excel and the instantaneous probability of being in any given state quickly converges to 1/2 as you approach x=2, so you can think of it as being in both states at once since it’s no long a value but rather a probability distribution or superposition. If the Grandi series equaled 1, it was be on, if 0 it would be off. It’s halfway between those two possibilities (1/2) hence in superposition.

A quantum-mechanical superposition is not a probability, you need to square it to get probabilities. Your reasoning is as valid or invalid as anybody else's though. For example, one could define the probability of 0 and 1 in a finite interval as you approach t-limit, e.g. between t-limit - x and t-limit - x/2, with x sliding towards zero. I believe that this will simply alternate between 0.25 and 0.75 and not converge nicely. That's the difference between convergent and divergent series: The convergent ones always give the same result regardless of method, the divergent ones don't.

@bob this series is different from Grandi series! Grandi series sums up to 1/2 (or -1/2 if you do not count the initial state), this series sums up to 0!

@toolforger the series definitely has no limit, but we can sum it up. Definitely, this series has a different sum than Grandi series.

@Annix I haven’t confirmed it myself last a cursory look but the Wikipedia page for Thomson’s lamp appears to show its connection to the Grandi series.

@bob one can draw some parallels, but that series and the one in the paradox have different sums.

@Anixx the sum of a divergent series is a meaningless concept, particularly for alternating series: The sum can be arbitrarily large if you can reorder the terms, or even diverge e.g. if you reorder it as 1+1-1+1+1-1+1+1-1+... I.e. the "sum" of the Grandi series does not exist, so you cannot draw conclusions from it.

@toolforger so do not reorder the terms. This problem in its formulation already has the order. The sum (for instance, Cesaro) of Grandi series exists, and it is 1/2.

@Anixx I am just advocating against drawing conclusions from the "sum" of the Grandi series; this sum does not exist. You may get convergence to 1/2 if you inntegrate the graph of the Thomson lamp, but that is not the Grandi series but an exponentially declining series which does converge but I'm not sure if the convergence is meaningful.

@toolforger the sum does exist if you sum it up properly. I do not draw conclusions from that series as that series is not applicable to Tomson's lamp problem.

@Anixx there is no "proper" way to sum a Grandi series!

@toolforger, there are, Cesaro, Borel, Abel and any other will give 1/2. wolframalpha.com/… Also, the numerosity of moves up is 1/2 greater than the numerosity down.

That does not change that the Grandi series diverges. And if you mean cardinality by "numerosity", then the cardinalities of the positive and negative sums is exactly the same. Unless you've been talking about the limit of Thomson's Lamp integral, which might indeed be 0.5, but that's not the Grandi series.