[math]{\frac{1}{2}}+{\frac{1}{4}}+{\frac{1}{8}}+...+{\frac{1}{\infty}}=1[/math].

 

Now there is a theoretical lamp(Thomson's Lamp) which can instantly switch between being on and off. If it is on for half a minute, then off for a quarter of a minute, then on for an eighth of a minute, at EXACTLY one minute, is it on or off?

Fairly simple actually. The definition doesn't say what state it is in after a minute any more than it says what colour the lamp is or what Thompson had for breakfast when he came up with this.

 

The definition only covers the condition of the lamp for any given moment up to one minute, but not at one minute or any time afterwards.

Do you ever actually reach "exactly one minute," or aren't you always just approaching it, closer and closer with each switch, but never arriving? Since the ticks get closer and closer, the light would be flashing so quickly as to (at least) appear continually on, regardless if one minute is reachable.