This argument convinces many experts that ½ is the actual limit of the Grandi series. But what does this mean for the thought experiment with the Thomson lamp? At the end of the two minutes, is the room half lit and half dark? The state of the lamp can be described for every moment, however brief, before the period of time has expired. But at exactly two minutes, the result remains a mystery. To shed more light on this, the physics philosophers John Earman and John D. Norton have placed the thought experiment in a somewhat more real environment.
A reasonably realistic implementation of the thought experiment
Suppose a metal ball is dropped on an induction cooktop. First the ball is in the air for one minute, then 30 seconds, then only 15 and so on. She repeats it endlessly for two minutes, each time generating an electrical impulse in the record. This is connected to a lamp that lights up with every contact. Since the ball comes to a standstill after two minutes due to gravity on the metal plate, the lamp is switched on at the end. So the camp that claims the limit is one would be right.
But the situation can be reversed: if the impacting ball does not close the circuit between the lamp and the induction field, but opens it. In this case, the lamp lights up and turns off whenever the ball lands on the plate. So after the two minutes have elapsed, the lamp is off. According to this interpretation, the limit of the Grandi series is zero. Norton and Earman therefore conclude: “The Thomson lamp is not a paradox but a problem that is incompletely described.”
If one strictly follows the mathematical definition of the limit value of a series, then there is only one conclusion: there is no limit value, ie the Grandi series diverges. On the other hand, if you expand the concept (such as adding roots of negative numbers to the real numbers), most experts stick to ½ as the solution—although that doesn’t explain how Thomson’s thought experiment turned out.
And to add to the chaos, there are also some people who insist on a completely different limit, which is neither 0 nor 1 nor ½. Because there are also plausible calculations that suggest that the Grandi series against the value ⅔ converges. For example, one can look at the equation (1 + x)/(1+x + x2) = 1 − x2 + x3 − x5 + x6 − x8th + … view. If x = 1, the right hand side of the Grandi series corresponds to the value ⅓ and the left hand side.
This leaves the mystery of the Thomson lamp unsolved. To be on the safe side, one can argue that such a thought experiment can never be implemented: In the real world, neither a human nor a metal ball can ensure that a lamp actually lights up infinitely often within two minutes. Or you can simply listen to the warnings of the parents and leave the switching on and off altogether.