A Harmonic Series Paradox (TANTON Mathematics)

Here's a cute, but disturbing, geometric interpretation of two famous series in mathematics: the harmonic series (which sums to an infinite value) and the Basel series (which sums to a finite value). Put the two together and ... WHOA!

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• HarmonicParadox

[erdmat61]

The reason this does hold true is the fact that you are using a pen to draw the line. No matter how thin you draw the line, say the line has a thickness of X, you will still have X(1) + X(1/2) + X(1/3) + X(1/4) + ...Then factoring out we have:

X(1 + 1/2 + 1/3 +1/4 + 1/5 + ...). Then no matter what thickness you choose for X, you will still be multiplying it by an infinite sum. So there will reach a point where the line is actually thicker than the boxes.

[DrJamesTanton]

@erdmat61 Good thinking. But are you saying that you still feel we can paint all the boxes? Us humans trying to do infinite things!

[DrJamesTanton]

No matter what size you decide to make the first box, &call that side-length 1 unit, you will get to boxes smaller than an atom and then smaller & smaller still. The mathematics seems to be saying that the areas of these things will always add up to a finite amount & so the areas can all be painted (though it is humanly impossible to paint infinitely many things - we're playing a mind game) but the side lengths add to an inifinte length and can't be drawn! Weird! Is this addressing your concern?

[boobah1067]

Hi there. Thank you for the tutorial. Why do you say 1 + 1/2 then 1/2 + 1/2. Shouldn't you start # 2 with 1 + 1/2 +1/2 ...? in other words, you don't repeat your term 1 actually. (I think the paradox occurs because we're dealing with a new start point: the original 1 + ... represents one circle, and the 1/2 + 1/2 ... is a different sized circle. Yes, they're both infinite in possibility each, but also different relatively, so one is "greater" from a single point of view. Do you understand?)

[DrJamesTanton]

@boobah1067 G'Day! I am sorry that my handwriting is hard to understand at times, but the math I am doing is correct. Since 1 is bigger than 1/2 we certainly have that S = 1 + 1/2 + 1/3 + 1/4 + ... must be bigger than 1/2 + 1/2 + 1/3 + 1/4 + ..., and since 1/3 is bigger than 1/4, this in turn must be bigger than 1/2 + 1/2 + 1/4 + 1/4 + .... And so on. That is all I am doing. (This might be confusing b/c I am not doing the standard proof everyone sees comparing it to 1/2 + 1/2 + 1/2 + ....)

[DrJamesTanton]

I am afraid to say that I don't understand your comment. Each square is "thicker" than the portion of line that it sits on. So filling in an area uses more ink than drawing the think line. So ... how can the areas use a finite amount of ink, but the line an infinite amount?

[silvascientist]

Finite area, infinite length, sounds pretty normal to me. I mean, isn't that what fractals are all about?

[AlexMack85]

@erdmat61 The field axioms don't hold for infinite sums if they do not converge. In particular, the distributive law does not hold. So we can't write x(1 + 1/2 + 1/3 + ...) = x(1) + x(1/2) + x(1/3) + ...

[EclecticSceptic]

@ethandrood No I think your point about area and lines is good. The thing I pointed out is immaterial really.

[ethandrood]

@EclecticSceptic OK I see your point. I still think that the 'paint paradox' is flawed though because a line segment has no area and so cannot be painted.

[EclecticSceptic]

What the hell actually. I'm going to try and figure this out. This is so weird.

[EclecticSceptic]

@ethandrood Interesting. However something you said is incorrect. The harmonic series, if viewed as steps from one point to another, would not be 'infinitely decreasing' as you mean. The harmonic series diverges, therefore you would eventually pass the point you wish to travel to. This is unlike the geometric series of Achilles and the Tortoise, which converges to 2.

[ethandrood]

Furthermore an infinitely thin object (such as a line) would have to be infinitely long in order to manifest a length. This may explain why an infinite number of decreasing steps is needed (1/2+1/3+1/4+...) in order to move between 2 points on a line. (There are different kinds of infinity it seems?). In fact some lines can't exist, like a line of length SQRT(2). Lines are very strange!

[ethandrood]

I think the problem here is with the definitions of a line. A line has length only and no area so we cannot paint a line, we would need infinitely thin paint! Area and length have different dimensions. We cannot compare them. In reality a line doesn't exist.

[ThePrevenge]

@messakg123 This is because, eventually, the next shape you draw will be so small that it will not change any number already written in the total sum, it will just add another decimal. It's the same way if you try to write pi using our number system. If you start with something as simple as 3,14 you can then "make it bigger" by making it 3,141 and then 3,1415 and so on. Still, it will never become bigger than 3,15.

I believe you're right though, this probably has nothing to do with the paradox.