Limits of infinite series and lightbulbs

Factual Questions
1

I was just thinking about a question I read a long time ago in one of Martin Gardner’s books, and wondered if any mathematically inclined Dopers could shed some light on it.

If you have the following infinite series: 1+1/2+1/4+1/8+1/16+1/32+…, then the limit of the series as the numbers approach zero will be 2. I’m ok with that.

The question from the book, however, was as follows: if you have a lightbulb that is on for 1 second, off for 1/2 second, on for 1/4, off for 1/8, etc., at the 2 second mark, will the light be on or off? Can this question even be answered, as it boils down to whether infinity is even or odd?

My own question as I was thinking about this was: Although the duration of each on/off cycle is getting shorter and shorter, time is still moving ahead normally. What would an observer watching the bulb see after the two second mark has passed (in theory. In reality, I’d see a bulb that’s burned out from being flipped on and off too much)?

2

What happened to the answers that had accumulated to this thread?

3

Alas, lost when the board was restored to Monday’s backup - shame, there were some quality replies here.

I think we reached a consensus that we’re not talking about the limitations of tungsten filament technology and alternating current…

I’ll try to remember what it was I said; essentially, I think it was:
Eventually (before you reach 2 seconds), you run into the ‘graininess’ of the universe; you will be trying to switch less than a whole electron and emit less than a whole photon for less than a Planck interval.

The question of what happens after 2 seconds is meaningless because by that time you are adding intervals of 1/(‡+n).

4

Hmm… could this be similar to the Schrodinger’s Cat, where the result is that the lightbulb is both on and off at the same time… or half-on, half-off?

5

I was going to say something very similar; that conceptually, infinity could be both even and odd at the same time; it makes about as much sense as any other attempt to treat it as a number.

6

Dammit that is supposed to read:
1/([symbol]¥[/symbol]+n)

7

**Please forgive me for repeating a post that disappeared earlier this week… **

For the physics and math folks: could the answer to the OP be tied up in whether space and time are quantized? If there is a minimum (quantum) increment of time, shouldn’t that provide the answer because at a certain point, the light bulb would not be able to switch back and forth any faster? Similarly, quantized space/time could provide a solution to Zeno’s Paradox by describing motion as a series of instantaneous jumps across the minimum space quantum, followed by a “resting” time interval of a certain duration, followed by another instantaneous jump, etc. The speed of light in a vacuum could thus be described as a photon performing this type of motion, each “resting” period being the minimum length of time (one time quantum). The ratio of the space quantum to the time quantum would be ©.

Is this feasible?

9

“One over yen plus n” - what’s that supposed to mean?

10

That’s supposed to be an infinity symbol.

From a purely mathematical standpoint, the lightbulb is neither on nor of at 2 seconds. This is equivalent to asking what the limit of the sequence 0, 1, 0, 1, … is, and that’s divergent.

11

Do you not have the symbol font on your system?

12

To answer this question, do the experiment.

What’s that? The experiment can’t be done? Then what’s the sense of asking the question?

13

Mangetout, the Symbol Font does not work in Netscape. Nor, I have found, does it work in Mozilla. You should use write it like this instead:

1/(infinity + n)

14

Mangetout - I have the symbol font, but I also get the Yen symbol under IE6.0, Windows 2000. Face it, anything that’s not 7-bit ASCII is going to have a problem somewhere :wink:

Chronos - I feel you’re being a little harsh on the OP. A lot of useful insights can be gained from thought experiments - look at how much productive discussion the theoretical EPR experiment has led to. (I think I read somewhere that we are now at the point where we can actually perform that experiment.)

Regarding the OP - this topic is covered a little in Rudy Rucker’s “Infinity and the Mind”. The theoertical device he describes is called a “Thompson Lamp”, and the related series 1 - 1 + 1 - 1 + … is called the Grandi Series. Rucker shows that depending on how you bracket the terms, the “sum” of the series could be considered 1 or 0:

(1 - 1) + (1 - 1) + (1 - 1) … = 0

-or-

1 + (-1 + 1) + (-1 + 1) … = 1

His lamp starts out off, and at noon every day the string is pulled to toggle it, *ad infinitum[/a] He asks whether the lamp will be on or off after an infinite number of days.

Here’s his conclusion: We can’t say whether the lamp will be on or off - knowing its behavior after a finite number of days does not allow us to extrapolate to an infinte number of days. He also gives seemingly equally valid reasonings for the lamp being on and off after infinitely many days.

15

Let’s say someone gave you the series:

1 + 1/2 + 1/4 + 1/8…

You and this person agree that the sum approaches two. He now asks you “what fraction are you adding when the sum is at 3?”

This doesn’t make any sense of course. And it doesn’t make sense at 2 either. Same for the light bulbs.

16

I’m shocked and surprised at hearing this from you Chronos there are a whole load of maths theories that cannot be physically realised; surely that isn’t the measure of whether something is worthy of interest and investigation…

17

I think Chronos was poking a hole in his cheek with his tongue when he made that post.

18

Ah, eat a bowl of whoosh.

19

There are two ways to approach the problem… One is theoretical in which case you’ll never reach two seconds…

The other is practical in which case you wear out your hand from flipping the switch so damn quickly.

20

I think I have a handle on this problem, but what does that mean, you’ll never reach two seconds?

21

Let me just say that as a professional mathematician, I second Ultrafilter’s answer. This is assuming that the physics of the universe would allow such a thing, which it wouldn’t. Even if space and time are not quantized (no graininess), no switch could carry out the necessary halving of the intervals.

You’ll reach 2 seconds sure enough, but you will not carry out the instructions.

Here is an even stranger paradox, which just shows that theoretical computations have their limit. If you imagine a ladder leaning against a wall and then sliding down the wall in such a way that the top of ladder falls under gravity (or even at a constant speed) you can calculate that at the time the top of the ladder hits the ground, the bottom of the ladder is moving away from the wall at infinite speed! What you can safely conclude is that it is not possible for the ladder to slide in such a way that top of the ladder slides at constant speed.