Something drops half a distance

Originally posted by DeepThought
It also means the sum has the property that doubling each element will increase the sum by two.

Easy to state, not so easy to prove. You will notice that Wikipedia does not actually prove that the sum of the series is the upper bound, but rather defines it as such. Only once such a definition is in place, can your statement can be proven.

Good grief!
I've got it.

I want to thank you all for the help.
It's perception!

The problem is that I've been looking at the paradox as: The glass leaves my hand and starts moving (and where it's moving to, in this case the floor, isn't important).
So, the glass leaves the hand and starts moving a distance, but first half that distance, then half a distance again.
The distance can't be breached.

The way to look at it, though, is: you have my hand and you have the floor. That's a set distance. For example: 1 meter.
Now, I can split that meter in two: 2x 50cm. If I add 50cm and 50 cm, I get the full meter.
Now, I can split that meter in an infinite amount of distances. If I add all those distances together, I get the full meter.

So the amount of divisions within that meter doesn't matter. If I add all those divisions up, I'll get a meter.

Originally posted by twhitehead
Easy to state, not so easy to prove. You will notice that Wikipedia does not actually prove that the sum of the series is the upper bound, but rather defines it as such. Only once such a definition is in place, can your statement can be proven.

That's because the sum of a convergent series is defined to be the limit of the sequence of partial sums. If the limit exists then the series is convergent and and the sum is the limit. We've shown that the partial sums converge so algebraic operations are safe on the series.

Originally posted by DeepThought
That's because the sum of a convergent series is defined to be the limit of the sequence of partial sums. If the limit exists then the series is convergent and and the sum is the limit. We've shown that the partial sums converge so algebraic operations are safe on the series.

I agree. I just thought it important to point out that the sum is defined as such and is not an actual arithmetic sum of all terms.

Originally posted by DeepThought
The series 1 + 1/2 + 1/4 + ... + (1/2)^N + ... is absolutely convergent because you can write the nth partial sum as 2 - (1/2)^(N+1), so the sequence of partial sums is bounded from above by 2. This means you can reorder the sum to your hearts content and it will not change the result of the sum. It also means the sum has the property that doubling each element will increase the sum by two.

Just a few minor quibbles:

The series 1 + 1/2 + 1/4 + ... + (1/2)^N + ... is absolutely convergent because you can write the nth partial sum as 2 - (1/2)^(N+1), so the sequence of partial sums is bounded from above by 2

The first minor quibble with this is that absolute convergence is defined on the absolute values of the terms of a series.
The second minor quibble is that for a series to be convergent it isn't sufficient for it to be bounded above. The partial sums have to be increasing. Since all of the terms of this series are positive talking about the terms or its absolute values is equivalent and the partial sums are increasing.
Hence one can conclude that the series is convergent (increasing partial sums that are bounded above) and that is also is trivially absolutely convergent (the terms and their absolute values are always the same)

This means you can reorder the sum to your hearts content and it will not change the result of the sum

This is only true for absolutely convergent series. And I'm aware that you stated that the series under consideration is absolutely convergent but you didn't provide an argument for it to be so.

These quibbles of mine are only in the interest of completeness because given the content of your posts it is evident that at the very least you have an intuition about all of I just talked about,. but maybe some people that read your posts don't have it and it's nice to let things be clear as they can be.

This is all well and good but who's going to clean up all of that broken glass on the floor?
Without going into a lot of mind numbing detail:

.999999... < X < 1

If you can find a number value for X you have a paradox; if no number value for X exists then there is no paradox.

easy peasy lemon squeezy


Reveal Hidden Content

Originally posted by lemon lime
This is all well and good but who's going to clean up all of that broken glass on the floor?
Without going into a lot of mind numbing detail:

.999999... < X < 1

If you can find a number value for X you have a paradox; if no number value for X exists then there is no paradox.

easy peasy lemon squeezy


[hidden]there is no paradox, the glass will hit the floor[/hidden]

Even if we are able to visualize the glass traveling along an infinite number of half distances, the glass doesn't actually travel forever along an infinitely long path. This 'paradox' relies on getting you to accept an imaginary number (infinity) and applying it to a real situation... it's like a magic trick that takes place in your mind, it asks you to take an imaginary number that is not a real number and apply that to a real (distance) number.

No value for X can be found for .9999... < X < 1. This means .9999... = 1, because there is no number to be found between those two values. So the glass is actually traveling along a set measured distance and not an infinite distance or infinite sequence of half distances. In spite of being able to image an infinite number of half distances, the glass is not traveling forever along a path of infinite distance.

Originally posted by lemon lime
Even if we are able to visualize the glass traveling along an infinite number of half distances, the glass doesn't actually travel forever along an infinitely long path. This 'paradox' relies on getting you to accept an imaginary number (infinity) and applying it to a real situation... it's like a magic trick that takes place in your mind, it asks you to ...[text shortened]... , the glass is not traveling forever along a path of infinite distance or sequence of distances.

By the same token (and as deepthought has already pointed out) adding up an infinite number of half distances will equal the whole distance.

Originally posted by shavixmir
Good grief!
I've got it.

I want to thank you all for the help.
It's perception!

The problem is that I've been looking at the paradox as: The glass leaves my hand and starts moving (and where it's moving to, in this case the floor, isn't important).
So, the glass leaves the hand and starts moving a distance, but first half that distance, then half ...[text shortened]... f divisions within that meter doesn't matter. If I add all those divisions up, I'll get a meter.

I keep forgetting to read all the way to the end before posting anything... I just now read your last post. You got it.

I still maintain that this so called 'paradox' is a bit of a magic trick that can be played on the mind.

Originally posted by lemon lime
This is all well and good but who's going to clean up all of that broken glass on the floor?
Without going into a lot of mind numbing detail:

.999999... < X < 1

If you can find a number value for X you have a paradox; if no number value for X exists then there is no paradox.

easy peasy lemon squeezy


[hidden]there is no paradox, the glass will hit the floor[/hidden]

X=(.999999...+1)/2 😏😏😏

😵😵😵

Originally posted by adam warlock
X=(.999999...+1)/2 😏😏😏

😵😵😵

But the answer is not such that .999999... < X < 1 ?

Originally posted by twhitehead
But the answer is not such that .999999... < X < 1 ?

It was in jest because I know that 0.99999...=1 hence there can be no X that is strictly between 0.99999...and 1. But if the numbers "a" and "b" aren't equal and, a<b, you can always define a number that is strictly between them (strictly speaking this holds for real numbers) by c=(a+b)/2. In fact this number is right in the middle of them

a < (a+b)/2 < b

Originally posted by adam warlock
X=(.999999...+1)/2 😏😏😏

😵😵😵

I looked up the word paradox to see if this really is a paradox or not. Apparently it is. Anything that appears to be a contradiction but isn't actually a contradiction can be a paradox. A good paradox fools the mind into thinking you're looking at a true contradiction... but I'm not sure if there really is anything that can be called a 'true' contradiction.

Originally posted by lemon lime
I looked up the word paradox
... but I'm not sure if there really is anything that can be called a 'true' contradiction.

I looked up the word "idiot".

Just out of curiosity.