Stumbling Intuition #3: Summing to Infinity

[Debater Slayer]

Previously, I made two threads discussing two subjects in mathematics that may be counterintuitive to some people:

Stumbling Intuition #1: The Monty Hall Problem

Stumbling Intuition #2: 0.999999 (Ad Infinitum)

For the third thread in the same series (no pun intended), I have decided to discuss the sum of a series to infinity. A related but not equivalent concept is one of Zeno's paradoxes of motion, simplified (arguably overly so) thus:

1. Dichotomy paradox: Before an object can travel a given distance d, it must travel a distance d/2. In order to travel d/2, it must travel d/4, etc. Since this sequence goes on forever, it therefore appears that the distance d cannot be traveled.

Zeno's Paradoxes -- from Wolfram MathWorld

A more detailed version can be found here:

Zeno’s Paradoxes (Stanford Encyclopedia of Philosophy)

To visualize the above (ostensible) paradox, here's an image depicting the infinite halving of distance that occurs when a person or object travels a distance d:

Per the above wording, the intuition behind the "paradox" is that an infinitely divisible distance d can never be covered within a finite amount of time. This is where two concepts come in: convergence of series and the sum of a geometric series to infinity.

To start with, a series is typically defined in mathematics as the addition of the terms of a given sequence of numbers. So, for example, the sequence we have above is d, d/2, d/4, d/8, d/16, ..., d/n, where n may be an infinitesimally small number. When we add these distances together, we have a geometric series.

A geometric series is one where there's a constant ratio between each term and the next. On the other hand, an arithmetic series is one where the difference between the consecutive terms is constant, as in 1 + 2 + 3 + 4 + ... + n, where the difference between each term and the one following it is 1.

In this case, common ratio of the geometric series is 1/2: each term is half the one preceding it. And since this common ratio lies between -1 and 1—so that its absolute value can never exceed 1—the series converges: we know that it will keep getting smaller ad infinitum. Each term is smaller than the preceding one, not greater.

This leaves us with the second concept, which is the sum of a series to infinity, in this case a geometric series. Zeno's paradox, as worded above (again, arguably in an oversimplified manner), assumes that an infinitely divisible distance can never be covered within a finite amount of time.

However, because the series converges, the terms become negligibly small as the number of terms approaches infinity. That is, as n approaches infinity in 1/2 raised to the power of n, the sum of Zeno's series can be calculated using the mathematical concepts of limits and series, and it comes out to a finite value:

So, while it may be intuitive to think that infinitely divisible quantities can never sum to a finite value, this is incorrect, because some of them can. Think of how you can theoretically divide a pie in half, and then in quarter, and then in an eighth, continuing ad infinitum until you have infinitesimally small pieces of pie. This division, assuming you have a sufficiently tiny knife, can continue infinitely, but the pie is still a finite quantity and indeed sums to one.

The last thing I want to note is that Zeno's paradoxes, including the one I used here, have been misrepresented and worded incorrectly in a lot of sources, so I don't claim that the version I cited here covers all the nuances of his dichotomy paradox or necessarily reflects the exact way he formulated it. I just used a popular formulation in order to demonstrate the concepts of series and infinite sums.

 

[Misunderstood]

Saw your post on being made an Admin on a Math Server. Congratulations, that is a very great achievement. I always liked math, it is so logical and elegant. So always glad to see someone into math and doing something with it. Just making me a bit envious.

But this is where math can get in the way of somebody actually trying to do something. A guy wants to take 1 step (d= one step), he is told to wait a bit because I want to do the math. The guy wanting to take the step asks how long he must wait? He is told he will need to wait infinitely, because the problem will go into infinity.

On a different note. It sounds like you answer questions. I was wondering what the value of zero is?

 

[Heyo]

But this is where math can get in the way of somebody actually trying to do something. A guy wants to take 1 step (d= one step), he is told to wait a bit because I want to do the math. The guy wanting to take the step asks how long he must wait? He is told he will need to wait infinitely, because the problem will go into infinity.

You don't have to wait infinitely, that is the (paradoxical) solution to Zeno's paradoxes from calculus.
In fact you can exactly calculate how long it will take to make the step.
Let's say it takes one second to make a full step. Then it would take half a second to make half a step, a quarter of a second to make a quarter step etc.
Some infinities go away when you subtract them.
The same applies to the summation. When I can sum up 1/2 and 1/4 in half a second and need always half the time for the next addition, I'd ask the one about to make a step to "wait a second" while I do the maths.

 

[Heyo]

On a different note. It sounds like you answer questions. I was wondering what the value of zero is?

The value of zero (0) is zero (0).

But I guess that is not your real question. You may have a misunderstanding of "zero". Zero is just a number. But you might be on to something. Contemplate the solutions to these equations:

a) x² = 0
b) x² + 1 = 0

 

[Misunderstood]

Very good, but the guy waiting to take a step still needed to wait until Calculus was developed for the mathematician to tell him he could take the step because he can calculate it now.

Or the guy waiting to take a step, could take his step and another and another until he gets to where he wants to go.

Just having some fun, not much into Paradoxes, but it does kind of prove that that infinity is not forever.

 

[Misunderstood]

See you get the Zero paradox too.

If 0 is zero, and zero is nothing; how can it be nothing, it has a name and a symbol. How can something with a name be nothing.

See if I write 0 0 or 1 2. They are something, the blank space in between is nothing. 1 space 2 would confuse people, they would not know what it means as there is no operator or number between them. Maybe it is the start of a list 1 2 3 4 5 6 7 8 9, who knows.

If I used some other symbols you may think there are three people, so the number is three. Like 1 2 you may think I am representing 3 things and forgot the plus sign.

But wait a minute, if I don't put a space between them and use the symbol zero for nothing like this 102; I get 102. 1 and 2 is 3, how did zero become 99?

I'm confused, I guess I need to go back to first grade and learn the basics over again. I think I missed something.

 

[Heyo]

If 0 is zero, and zero is nothing;

That's the misconception. Zero is not nothing. Is 0° the absence of temperature?
When you draw the graphs for x² and x²+1, the connection of the y-axis with the first graph gives you 0, a real solution, whereas the second graph has no such intersection, i.e. the solution (in real numbers) is an empty set {}, not 0.