Thompson's Lamp

At some point the switching rate will have to exceed the speed of light, but can't. So the last state (off or on) will be the last state before that limitation is reached. A simple computer program or some clever mathematics could provide a definite answer. Any suggestions?

These sort of problems are called gedankenexperimenten , or thought experiments , and they have been around since the beginning of logical reasoning. Usually the context in which they are used gives a sort of list of parameters to consider in order to solve them or even just try to understand them and/or see if they lead to contradictions. In this case we are not dealing with a physics gedankenexperiment, which would undoubtfully take physical constants as the speed of light or theories such as the Special and General Relativity into account, but with a purely mathemathical one about the convergence of infinite series, and what something like an infinite sum of infinitesimal quantitities adding up to a finite value means, or even just if it makes sense to consider such a thing possible.

So it makes no sense to bother about the real physical world in this case, and things such as the power of the emitting source, the wavelenght and/or frequency of the electromagnetic radiation emitted (visible spectrum?) or even the speed of light are irrelevant and shouldn't be taken into account. I have read comments about people arguing that the light bulb would burn out because of thermal stress, but why should one assume that because we are talking about a lamp then it must have a light bulb? Couldn't it be a LED lamp? Or even just a shutter that blocks/permits the sunlight to enter into a dark room through a small hole? But wait, what if there where two holes and not just one? Then there would be diffraction and interference! And what if the man making this experiment is blind? How could he tell if the light will be on or off at the end? Must it be a human necessarily? We could go on with these sort of irrelevant questions forever, but we would be missing completely the point of this gedankenexperiment.

Anyway, I will still try to answer your question. First, to compute what you call the physical limit imposed by the constraint of the speed of light as the maximum switching rate achievable, you could just measure the lenght of the path of the part of the switch that travels the longest distance (let's assume that it would be its tip, and the path to be a circular arc of radius $r=3\text{ cm}$ , maybe a quarter of a circle to keep the calculations easy ). So we have:

$L_{path}=\frac{2\pi r}{4}=\frac{\pi}{2}3\text{ cm}\approx4.7124\text{ cm}=4.7124\times10^{-2}\text{ m}.$

Considering the speed of light to be

$c\approx300000\frac{\text{ Km}}{\text{ s}}=3\times10^8\frac{\text{ m}}{\text{ s}}$

the limit time would be

$t_{\text{ limit}}=\frac{L_{\text{path}}}{c}=\frac{4.7124\times10^{-2}\text{ m}}{3\times10^8\frac{\text{ m}}{\text{ s}}}\approx1.5708\times10^{-10}\text{ s}.$

Now, considering we start with the light on , and the first interval had a duration of $60\text{ s}$ , because at every switch we are halving the duration of the next interval, the duration of the $n$ -th interval will be

$t_{n}=\frac{60\text{ s}}{2^n}.$

Imposing $t_{n}=1.5708\times10^{-10}\text{ s}$ and solving for $n$ yields

$n=\frac{log\frac{60\text{ s}}{1.5708\times10^{-10}\text{ s}}}{log2}\approx551067624480$

and because $n$ is even , and we started with the light on , not changing the parity of the state will result in the light being still ON .

Ok, now that we have this nonsense answer, let's make some important considerations. I have used a lot of approximations along my calculations (some which were compulsory, as the approximation of $\pi$ , others because of constraints about our knowledge of the exact value of any physical constant (are they really constant , BTW?), and many others because of the limits imposed by the maximum number of binary digits that the (scientific!) calculator on my old phone is able to use in the calculations, not to mention wrong roundings caused by overflow.

Anyway, beside of all of these technical issues, I was able to determine the final state of the lamp. But know start the real big questions (which I admit I have no idea how to correctly debate!): Special Relativity imposes the speed of light in a vacuum as the limit speed of any phisical phenomenon involving the exchange of subatomic particles, but in the entire universe there are other phenomena that aren't necessarily limited by it (for example, the Universe itself expands at a much greater speed than that of light!). So what if our switch was somehow special and could be moved faster than the speed of light? Maybe we would be in the dark when the lamp is on and viceversa, because the light emitter receives the impulse to switch its state in a time shorter than what it takes for the light it emits to travel to our eyes! Also, when an object (in this case the switch) approaches the speed of light, as General Relativity states, its mass increases: at some point not even the strongest man in the world could move the switch (especially from the down-state to the up-state, against gravity!). Really, we could go on forever!

Please believe me if I say I didn't mean to be sarcastic with this comment (I even tried to figure out the final state of the lamp based on your assumptions); my only purpose was to clarify the importance that we need to give to the right details, especially in gedankenexperimenten, or the whole point they were invented for gets lost into useless considerations about improbable/impossible/unspecified situations.