After switching a lamp on and off infinitely many times in one minute, is it on or off? So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another quarter of a minute passes you turn it on and so on.
Now this process will take : $1/2+1/4+1/16+...=1$ minute
and this is the states of the lamp put into a sequence:
$(1,-1,1,-1,1,...)$
After a minute of switching it on an off passes, will it be on or off?
I'm well aware that the limit of this sequence does not exist, however, my intuition dictates that after a minute passes, it is either on or off. 
So does this lamp have a final state(on or off) which is impossible for us to know? or it does not have a finals state(it seems to me, it must have one!)?
 A: This is known as Thomson's lamp. The Wikipedia article is probably less enlightening than the answers given here, and I don't think that there's much more to be said about it, but the search term will lead you to more articles if you want them.
Added:  Your reasoning is correct. By definition, the lamp must be in one of two states: on or off. This state is observable and definite at all times. However, the state at one minute from the start (and hence at all later times) cannot be determined from the description of the lamp and switching process. So the "supertask" of this process does not determine an outcome. It was this feature that led Thomson to reject the idea of "supertask": it is no better than a random or arbitrary action.
Thomson's lamp is intended as an abstraction, like a Turing machine. Considering it as a real physical system does not follow Thomson's formulation; rather, it addresses an altogether different conception. Also, in my view, very little mathematics is involved in understanding this matter.
Further edit For what it's worth, here is a mathematical interpretation of the Thomson lamp problem. Define (!) $f:[0\;,\infty)\to\{0,1\}$ by $$f(x)=\begin{cases}\;\;1 & \text{if $1-2^{1-n}\leqslant x<1-2^{-n}$ for some odd $n\in \Bbb N$,} \\ \;\;0 & \text{if $1-2^{1-n}\leqslant x<1-2^{-n}$ for some even $n\in\Bbb N$,}\\f(1) & \text{if $x\geqslant1$.}\end{cases}$$The Thomson lamp problem is then "What can we say about $f(x)$ for $x\geqslant1$?". The answer is nothing---because $f$ has not been defined in this range, since $f(1)$ has not been defined. In fact, the same answer would apply regardless of how $f$ had been defined on the range $[0\;,1)$.
A: 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.
A: Consider the sequence $a_n=1-2^{-n}$. The situation that you describe could be mathematically described this way:

For each $x\in[0,1)$ there exists some unique $n\in\Bbb N$ such that $x\in[a_n,a_{n+1})$. Define $f(x)=1$ if this $n$ is even and $f(x)=0$ otherwise.

As you can see, the value $f(1)$ remains undefined.
So, assuming that the job of toggling infinitely many times the lamp switch is physically possible, there would be no way to say the lamp state after one minute. 
A: The apparent paradox in this question arises from the conflict between the following two ideas:


*

*Our intuition about physical lamps.  We expect them to have certain properties, such as being on or off, but not both or neither.

*A complicated function that oscillates between two values at an increasing pace, whose oscillations become arbitrarily rapid.
It is not possible to have a physical lamp whose on/off state follows the complicated function.  This is the resolution to the paradox.  Thus we must either give up (1) or (2).  If we give up (1), then there is no reason to believe that the state must be either on or off.  The mathematical function has no limit.
A: To answer that question, you must first answer the question Is Infinity Even or Odd?
Assume the lamp is initially on. After one minute, if you flip the switch an even number of times, the lamp will be on. However, if after that minute, you had flipped the switch an odd number of times, the lamp will be off.
So if you flip the switch an infinite number of times, whether or not the lamp is ultimately on, depends on whether "infinity" is either even or odd. See the linked question above for the answer.
Basically, as it stands, the "infinity" you speak of is not really a number, so the question is ill defined, and therefore has no physical solution of "on" or "off".
A: The lamp is ill-defined.  You've given us a very thorough description of what happens before the minute is up.  But you've not told us what happens next, so the best we can do is guess.  We could imagine a lamp which is on after the minute ends, and we could equally well imagine a lamp which is off.  Both possibilities are compatible with all of the information you've provided.
A: Assuming the wiring is 100% efficient and the speed of the electrical current is instantaneous (which is just as illogical as the idea of unreal numbers such as infinite), there would be no final flip, as the time it takes to switch it is always divided by half.
However, in a real-world application, where wiring has latency and electrical current has a set speed, when the time between switches gets low enough that the switch is being turned on then off before any current could pass the wires (assuming any force could act that fast), the overall state will always be off.
