Concerning Infinity (of course)

[Boltzmannbrain]

My new issue in my journey to try to understand infinity concerns the "ends" of infinity.

I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1.  

I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1.  What could n possible equal to get to this point?

If this last point really is a solution to the equation, then wouldn't it have to be 1/infinity, or in  other words, the "infinity-ith" point?  If so, how can it be said that the natural numbers can numerate all points of a set of size aleph-null? 

Edited September 3, 2023 by Boltzmannbrain

[Genady]

  On 9/3/2023 at 8:46 PM, Boltzmannbrain said:

I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1.  

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By definition, it means the following:

for every real number d>0 there exists such natural number N that

|(sum from 1 to m of 1/2^n) - 1| < d

for any m>N.

Edited September 3, 2023 by Genady

[Genady]

  On 9/3/2023 at 8:46 PM, Boltzmannbrain said:

What could n possible equal

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n does not have a definite value in this expression. It runs from the lower bound of summation to the upper bound. 

  On 9/3/2023 at 8:46 PM, Boltzmannbrain said:

If this last point really is a solution to the equation

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There is no last point. There is no solution to the equation.

[Boltzmannbrain]

  On 9/3/2023 at 9:34 PM, Genady said:

By definition, it means the following:

for every real number d>0 there exists such natural number N that

|(sum from 1 to m of 1/2^n) - 1| < d

for any m>N.

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This is just the limit, right?  I want to know if the sum of all n fractions actually equals 1, but I see on your next response that you answered what I wanted to know.  Thanks.

The only thing is that I was told on a different forum that there is a solution, being 1.  They seemed quite knowledgeable too.  

Hmm, it's not letting me put the link to the other forum.

If you go to mathforums dot com and go to real analysis, scroll down about 21 threads to my thread (from mathmath) called "How close to 1 does this infinite sum get".  They seem to be agreeing that the sum does actually equal 1.

Edited September 4, 2023 by Boltzmannbrain

[wtf]

  On 9/4/2023 at 2:12 AM, Boltzmannbrain said:

This is just the limit, right?  I want to know if the sum of all n fractions actually equals 1,

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The sum is defined as the limit of partial sums. 

So if we have 1/2 + 1/4 + 1/8 + 1/16 + ..., the partial sums are:

1/2, 3/4, 7/8, 15/16, ...

The limit of the sequence of partial sums is 1. So by definition the sum of the original infinite sum is 1. 

It's explained here on Wiki.

https://en.wikipedia.org/wiki/Series_(mathematics)

"it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series."

 

Edited September 4, 2023 by wtf

[Boltzmannbrain]

  On 9/4/2023 at 2:23 AM, wtf said:

The sum is defined as the limit of partial sums. 

So if we have 1/2 + 1/4 + 1/8 + 1/16 + ..., the partial sums are:

1/2, 3/4, 7/8, 15/16, ...

The limit of the sequence of partial sums is 1. So by definition the sum of the original infinite sum is 1. 

It's explained here on Wiki.

https://en.wikipedia.org/wiki/Series_(mathematics)

"it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series."

 

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I could not find the part that you quoted in the link that you gave me.  But I did read in the same Wiki link (under the heading "Convergent Series") that the infinite sum of 1 + 1/2^n = 2.  They don't seem to be saying it is just a definition either.  What is going on here?

[wtf]

  On 9/4/2023 at 5:23 AM, Boltzmannbrain said:

I could not find the part that you quoted in the link that you gave me.

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End of 4th paragraph.

[studiot]

  On 9/3/2023 at 8:46 PM, Boltzmannbrain said:

My new issue in my journey to try to understand infinity concerns the "ends" of infinity.

I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1.  

 

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Yes you are considering the right things this time.

As regards the start point, even simple counting has a start point 1,2,3........

But it has no end point unless you run out of numbers to count., which of course you will not.

That is what is meant by infinity in this case.  (remember there are other cases)

 

  On 9/4/2023 at 2:23 AM, wtf said:

The sum is defined as the limit of partial sums. 

So if we have 1/2 + 1/4 + 1/8 + 1/16 + ..., the partial sums are:

1/2, 3/4, 7/8, 15/16, ...

The limit of the sequence of partial sums is 1. So by definition the sum of the original infinite sum is 1. 

It's explained here on Wiki.

https://en.wikipedia.org/wiki/Series_(mathematics)

"it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series."

 

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An absolutely spot on answer to the series part of the question. Clear compact and brings out the essential points.  +1

 

I would add to this that for further information @Boltzmannbrain  should study Cauchy sequences as this is one good way of dealing with this subject.

https://en.wikipedia.org/wiki/Cauchy_sequence

[Genady]

  On 9/4/2023 at 2:12 AM, Boltzmannbrain said:

I was told on a different forum that there is a solution

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It is easy to prove that there is no solution. Here it is:

The sum of 1/2^n from 1 to m equals 1-1/2^m. If there is a solution, then

1-1/2^m = 1,

which means

1/2^m = 0.

But 1/2^m > 0 for any m.

Thus, there is no solution.

Edited September 4, 2023 by Genady

[Boltzmannbrain]

  On 9/4/2023 at 6:13 AM, wtf said:

End of 4th paragraph.

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Isn't the notation on the bottom saying that the limit is the total sum?  If it doesn't actually equal 1 like the notation suggests, then why is it even used in the first place?

[wtf]

  On 9/4/2023 at 4:02 PM, Boltzmannbrain said:

Isn't the notation on the bottom saying that the limit is the total sum?

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The notation says that the sum is defined as the limit of the partial sums.

 

  On 9/4/2023 at 4:02 PM, Boltzmannbrain said:

If it doesn't actually equal 1 like the notation suggests, then why is it even used in the first place?

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It's defined to be 1, the limit of the partial sums. 

That's actually the clever part of the definition. We can't make sense of "what is the sum after infinitely many operations?" or "Isn't there a tiny little bit left over?" and so forth. The limit definition avoids those problems by providing a precise definition of the sum of an infinite series.

Edited September 4, 2023 by wtf

[Genady]

Perhaps it might be clearer if instead of

they have said

[Boltzmannbrain]

  On 9/4/2023 at 4:28 PM, wtf said:

The notation says that the sum is defined as the limit of the partial sums.

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Ok, I agree, but isn't it also saying that the infinite sum of 1/2^n equals 1?

  Quote

 

It's defined to be 1, the limit of the partial sums. 

That's actually the clever part of the definition. We can't make sense of "what is the sum after infinitely many operations?" or "Isn't there a tiny little bit left over?" and so forth. The limit definition avoids those problems by providing a precise definition of the sum of an infinite series.

 

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Then getting back to my original issue, if the definition is correct, doesn't this mean that there is an end?  This end would seem to be at 1.    

  On 9/4/2023 at 4:57 PM, Genady said:

Perhaps it might be clearer if instead of

they have said

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Is it wrong?  Other sources that I am reading have the equal sign with only two bars too. 

Edited September 4, 2023 by Boltzmannbrain

[wtf]

  On 9/4/2023 at 8:48 PM, Boltzmannbrain said:

Ok, I agree, but isn't it also saying that the infinite sum of 1/2^n equals 1?

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Yes. The partial sums of 1/2 + 1/4 + 1/8 + ... are

1/2, 3/4, 7/8, ...

and the limit of that sequence of partial sums is 1. So the sum of the original series is 1 by definition. 

 

  On 9/4/2023 at 8:48 PM, Boltzmannbrain said:

Then getting back to my original issue, if the definition is correct, doesn't this mean that there is an end?  This end would seem to be at 1.    

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Definitions can't be correct or incorrect, only useful or not, insightful or not. 

The entire point of the formal definition is to bypass meaningless and unanswerable questions involving "the end." There are no answers to those kinds of speculations nor is there really any meaning in them.

Instead, we substitute the clever definition that allows us to formally prove that the sum of 1/2^n is 1, and we let everyone have their own private intuitions as long as we agree to use the formalism. 

 

 

Edited September 4, 2023 by wtf

[Genady]

  On 9/4/2023 at 8:48 PM, Boltzmannbrain said:

Is it wrong?  Other sources that I am reading have the equal sign with only two bars too. 

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It is not wrong. But three bars make it clear that it is a definition rather than an equation.

Edited September 4, 2023 by Genady

[Boltzmannbrain]

  On 9/4/2023 at 8:58 PM, wtf said:

Yes. The partial sums of 1/2 + 1/4 + 1/8 + ... are

1/2, 3/4, 7/8, ...

and the limit of that sequence of partial sums is 1. So the sum of the original series is 1 by definition. 

 

Definitions can't be correct or incorrect, only useful or not, insightful or not. 

The entire point of the formal definition is to bypass meaningless and unanswerable questions involving "the end." There are no answers to those kinds of speculations nor is there really any meaning in them.

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I am still confused. 

I would understand if the definition was something like an arbitrary symbol like they did with an imaginary number i.  That would make sense to me with what your saying because it wouldn't be a value that already has a definite meaning. 

You are calling it meaningless, but 1 has meaning.  This is why I am still stuck with the problem in my OP.

[wtf]

  On 9/4/2023 at 10:07 PM, Boltzmannbrain said:

I am still confused. 

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Step 1: The limit of the sequence 1/2, 3/4, 7/8/ 15/16, ... is 1

Step 2: The sum of an infinite series is defined as the limit of the sequence of partial sums.

Step 3: The infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... has the associated sequence of partial sums 1/2, 3/4, 7/8, 15/16, ...

Step 4. Therefore the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... is 1, by Step 2.

Which part of that logic is giving you trouble?

If you can focus on the logic of these steps you will understand the process. You can have private intuitions about "the end" or whatever your intuitions may be, but when doing math, you need to focus on the math itself.

[Boltzmannbrain]

  On 9/4/2023 at 11:33 PM, wtf said:

Step 1: The limit of the sequence 1/2, 3/4, 7/8/ 15/16, ... is 1

Step 2: The sum of an infinite series is defined as the limit of the sequence of partial sums.

Step 3: The infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... has the associated sequence of partial sums 1/2, 3/4, 7/8, 15/16, ...

Step 4. Therefore the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... is 1, by Step 2.

Which part of that logic is giving you trouble?

If you can focus on the logic of these steps you will understand the process. You can have private intuitions about "the end" or whatever your intuitions may be, but when doing math, you need to focus on the math itself.

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I think I am just missing the point of the partial sums part.  I don't really know the point of it.  Why is it part of the process?  

[wtf]

  On 9/5/2023 at 1:56 AM, Boltzmannbrain said:

I think I am just missing the point of the partial sums part.  I don't really know the point of it.  Why is it part of the process?  

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Because that is how they define the sum of an infinite series.

We want to define the sum 1/2 + 1/4 + 1/8 + ...

But we don't know how to add up infinitely many things. 

So we DEFINE the sum to be the limit of the sequence of partial sums.

The sequence of partial sums is 1/2, 3/4, 7/8, 15/16, ... Can you see that?

Ok now the sequence of partial sums happens to have the limit 1. That follows from the definition of the limit of a sequence. Do we perhaps have to review that? That's actually the trickiest part of all this. Once we have that, the rest is easy.

The sequence of partial sums has the limit 1. We are not defining 1 differently, it's the same old familiar number 1. We are noting that the limit of the sequence of partial sums is 1.

Then, since our goal was to somehow define the infinite sum, we define the sum of the series 1/2 + 1/4 + 1/8 + ... to be whatever the limit of the sequence of partial sums is, which in this case happens to be 1.

Is that any more clear? Once we know what the limit of the sequence of partial sums is, we just make an arbitrary rule to define the sum of the series as the limit of the sequence of partial sums. Only in this case it turns out to not really be arbitrary, but actually rather clever ... since we've found a sensible way to define the sum of an infinite series. 

The reason it's sensible, is that the sum of that series should be 1 if it's anything at all; and now we've found a way to define it logically so that it works out. 

 

 

Edited September 5, 2023 by wtf

[Boltzmannbrain]

  On 9/5/2023 at 2:17 AM, wtf said:

Because that is how they define the sum of an infinite series.

We want to define the sum 1/2 + 1/4 + 1/8 + ...

But we don't know how to add up infinitely many things. 

So we DEFINE the sum to be the limit of the sequence of partial sums.

The sequence of partial sums is 1/2, 3/4, 7/8, 15/16, ... Can you see that?

Ok now the sequence of partial sums happens to have the limit 1. That follows from the definition of the limit of a sequence. Do we perhaps have to review that? 

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Do we have to use partial sums to define the limit of a sequence?  I don't remember leaning that.  I just learnt the definition E>0, n>N --> |an - L| > E

 

  Quote

 

The sequence of partial sums has the limit 1. We are not defining 1 differently, it's the same old familiar number 1. We are noting that the limit of the sequence of partial sums is 1.

Then, since our goal was to somehow define the infinite sum, we define the sum of the series 1/2 + 1/4 + 1/8 + ... to be whatever the limit of the sequence of partial sums is, which in this case happens to be 1.

Is that any more clear? Once we know what the limit of the sequence of partial sums is, we just make an arbitrary rule to define the sum of the series as the limit of the sequence of partial sums. Only in this case it turns out to not really be arbitrary, but actually rather clever ... since we've found a sensible way to define the sum of an infinite series. 

The reason it's sensible, is that the sum of that series should be 1 if it's anything at all; and now we've found a way to define it logically so that it works out. 

 

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I don't think that I am misunderstanding anything.  But maybe I am missing something important about the partial sum part.  I will explain what is going on in my head with a question.

The limit of the infinite series 1/2^n is 1.  Why isn't this enough to define the sum as 1?  Obviously the sum is 1 (or arbitrarily close).  Why do we need the partial sums part?  How does it help tell us that the series is 1?

[wtf]

  On 9/5/2023 at 2:46 AM, Boltzmannbrain said:

Do we have to use partial sums to define the limit of a sequence?  I don't remember leaning that.  I just learnt the definition E>0, n>N --> |an - L| > E

 

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No, we use the partial sums trick to define the sum of an infinite series.

As you note, the limit of a sequence is given by the epsilon-N definition. That's the right definition. 

(ps) The absolute value should be LESS THAN epsilon. 

 

  On 9/5/2023 at 2:46 AM, Boltzmannbrain said:

The limit of the infinite series 1/2^n is 1.  Why isn't this enough to define the sum as 1?  Obviously the sum is 1 (or arbitrarily close).  Why do we need the partial sums part?  How does it help tell us that the series is 1?

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Right, the sum of 1/2^n is "obviously" 1.

Now if someone says to us, well I don't believe that, how can you make MATH out of it? How can you lay down some basic principles from which the fact that the sum is 1 will follow LOGICALLY?

And THAT's what the limit of the partial sums definition is. It's a way of FORMALIZING what we already see must be true. 

In other words we already know what we want the answer to be. The partial sums are a clever way of constructing a framework in which what we "know" to be true, can be logically proven to be true. 

Does that help? Are we converging, no pun intended, to understanding?

The partial sums are a way of formalizing our intuition. Exactly in the same way that the epsilon-N definition formalized our intuition that the limit of the sequence 1/2, 3/4, 7/8, ... should be 1. We already know what answer we want to get. The epsilon-N definition is a formalization that lets us logically derive the answer we wanted to get in the first place.

In a sense, this whole business is an inversion of how people usually see math. People think that in math we have a problem and we want to find the answer. But in many cases, we already know the what the answer should be, and we want to create the math that gives us that answer!

 

Edited September 5, 2023 by wtf

[Boltzmannbrain]

  On 9/5/2023 at 2:55 AM, wtf said:

No, we use the partial sums trick to define the sum of an infinite series.

As you note, the limit of a sequence is given by the epsilon-N definition. That's the right definition. 

(ps) The absolute value should be LESS THAN epsilon. 

 

Right, the sum of 1/2^n is "obviously" 1.

Now if someone says to us, well I don't believe that, how can you make MATH out of it? How can you lay down some basic principles from which the fact that the sum is 1 will follow LOGICALLY?

And THAT's what the limit of the partial sums definition is. It's a way of FORMALIZING what we already see must be true. 

In other words we already know what we want the answer to be. The partial sums are a clever way of constructing a framework in which what we "know" to be true, can be logically proven to be true. 

Does that help? Are we converging, no pun intended, to understanding?

The partial sums are a way of formalizing our intuition. Exactly in the same way that the epsilon-N definition formalized our intuition that the limit of the sequence 1/2, 3/4, 7/8, ... should be 1. We already know what answer we want to get. The epsilon-N definition is a formalization that lets us logically derive the answer we wanted to get in the first place.

In a sense, this whole business is an inversion of how people usually see math. People think that in math we have a problem and we want to find the answer. But in many cases, we already know the what the answer should be, and we want to create the math that gives us that answer!

 

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Yes, the steps make sense now.  Thanks!

Now it seems like the argument has become more about whether or not infinite functions actually reach the limits they converge to.  So does something like 1/n ever reach 0 after using the whole set of natural numbers?

[wtf]

  On 9/5/2023 at 5:58 AM, Boltzmannbrain said:

Now it seems like the argument has become more about whether or not infinite functions actually reach the limits they converge to.  So does something like 1/n ever reach 0 after using the whole set of natural numbers?

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The purpose of the formalization is so that we don't have to think about meaningless questions like that. 1/n gets arbitrarily close to 0 as n gets large, that's the epsilon-N idea. So we define 0 as the limit of 1/n because it satisfies the epsilon-N condition. Then we don't have to confuse ourselves with unanswerable riddles like, "Does it ever reach 0?" Well actually it doesn't, since for any natural number n, 1/n is not 0. But it gets arbitrarily close. That's the idea of the limit concept. It avoids the meaningless questions. 

I don't know why you say it "seems like the argument has become whether infinite functions reach their limits." On the contrary, the formalization makes those kinds of questions irrelevant.

Edited September 5, 2023 by wtf

[Boltzmannbrain]

  On 9/5/2023 at 7:23 AM, wtf said:

I don't know why you say it "seems like the argument has become whether infinite functions reach their limits." On the contrary, the formalization makes those kinds of questions irrelevant.

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Because my issue in the OP has essential come down to the sequence from the partial sums of 1/2^n, namely (n-1)/n.  If this actually reaches 1, then my issue in the OP is still a problem.

+1's given yesterday

[studiot]

  On 9/5/2023 at 7:06 PM, Boltzmannbrain said:

Because my issue in the OP has essential come down to the sequence from the partial sums of 1/2^n, namely (n-1)/n.  If this actually reaches 1, then my issue in the OP is still a problem.

+1's given yesterday

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The question is not whether the sequences 'can reach 1' but whether you can.

Of course neither you nor any other being following along the sequence can ever get to the end.

 

I hope you agree that the numbers 1, 1.5 2, 22 million along with every other conceivable number, all existed ie were available to be discovered, long before Man evolved in the universe.

Luckily numbers take up zero space so there is plenty of room in our universe for all of them.

 

So the entire sequence was there before we were and has effectively always been there.