Some infinite series converge, and some do not. In this case, because of the halving of the interval between each switching event, the total time for an infinite number of such events does converge, and converges to a small number (1 minute if you start with a 30 second gap). However the state of the lamp is represented by a different infinite series which does not converge. As a result it does not have a 'final' state - and indeed several different 'final states' can be 'proved' by grouping the terms of the series in different ways. As a mathematical construct we have to discard our intuition about 'real' lamps or a 'final' value.
Alternatively remember that it IS a real lamp, and it and the switch are subject to real physical laws, including relativity and Quantum mechanics. Considering just the switch (and assuming it manages to avoid wearing out with repeated use), it needs to move a certain distance to break the circuit, and it needs to do so in less and less time - moving faster and faster until it approaches the speed of light and becomes infinitely heavy ... However hard you try, you can't then switch it back in half the time !
Avoiding that singularity, we have the effect of Quantum effects when our continued halving of the interval reaches the Planc time....and again the behaviour of our real Lamp diverges from that of our Mathematical construct.
But my final answer, is that the Lamp would be ON (unless you broke it). In attempting to switch it repeatedly, you could only achieve that if the position of the switch was itself a convergent series of smaller and smaller movements, in the shorter and shorter intervals. The Lamp would 'brown out' as the electricity was interrupted, but for shorter and shorter times, and the heat would cause the conducting parts of the switch to melt, and eventually fuse - when the current would flow, and the Lamp would be on. Stuck ON.