Trying to understand infinity (we are giving infinity a definite beginning point at 1/2 and a definite end point at 1)

[topsquark]

But then I am still confused about how the segments/terms reach the other side. A limit just gets arbitrarily close doesn't it? In other words, each n only gets the sum close. How does the gap get filled?

They never do!

The Nth partial sum is
$\displaystyle \sum_{n=1}^N \dfrac{1}{2^n}= \dfrac{1}{2} \left ( \dfrac{ 1 - \left ( \dfrac{1}{2} \right )^N }{ 1 - \dfrac{1}{2} } \right ) = 1 - \left ( \dfrac{1}{2} \right )^N$

Now, we want this in the limit as N goes to infinity. Well, nothing happens to the 1. What happens to the $(1/2)^N$? Notice that 0 < 1/2 < 1. Every time we put a higher exponent on this, the smaller it gets. If we do this an arbitrarily large number of times, we get arbitrarily closer to 0. So
$\displaystyle \lim_{N \to \infty} \left ( \dfrac{1}{2} \right )^N = 0$

Thus
$\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{2^n}= 1 - 0 = 1$

If you want to do the limit for real, look up epsilon-delta proofs. But the point is that we never took a value for N to find this. We simply noted that the bigger N is, the closer $(1/2)^N$ gets to 0. This is not the sort of process where you can imagine that you are summing an very large number of terms. We sum them for an arbitrary N and then we take a limit, which does the job for us.

-Dan

 

[Mates]

And by "it" you mean what? The limit? Then I agree. A partial sum? Then no, there are no partial sums equal to 1.

You said, "what would be the sum of the series? It would be 1."

Your "it" clearly referred to "sum", and not "partial sum". If you meant partial sum, then of course I agree.

 

[lev888]

You said, "what would be the sum of the series? It would be 1."

Your "it" clearly referred to "sum", and not "partial sum". If you meant partial sum, then of course I agree.

I am trying to avoid any miscommunication. So, you agree if I wrote that a partial sum equalled 1? I'll repeat, no partial sum equals 1. A partial sum is just a total of terms 1 through N. Post 21 clearly shows that it can't equal 1 for any N.
The infinite series sum is another matter. We can't actually total it term by term. But we _define_ its sum as the limit of the partial sum sequence (if it exists). It's not an actual sum. The gap never gets filled. How about this rule - every time you want to say "sum", if it refers to the sum of an infinite series you have to say "so-called sum". This may help you realize that not all sums are created equal.

 

[Mates]

They never do!

Then if it is not n that fills the gap, how does the gap get filled? There has to be something that is filling the gap. Please just tell me in simple terms exactly what is filling this gap.

The Nth partial sum is
$\displaystyle \sum_{n=1}^N \dfrac{1}{2^n}= \dfrac{1}{2} \left ( \dfrac{ 1 - \left ( \dfrac{1}{2} \right )^N }{ 1 - \dfrac{1}{2} } \right ) = 1 - \left ( \dfrac{1}{2} \right )^N$

Now, we want this in the limit as N goes to infinity. Well, nothing happens to the 1. What happens to the $(1/2)^N$? Notice that 0 < 1/2 < 1. Every time we put a higher exponent on this, the smaller it gets. If we do this an arbitrarily large number of times, we get arbitrarily closer to 0. So
$\displaystyle \lim_{N \to \infty} \left ( \dfrac{1}{2} \right )^N = 0$

Thus
$\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{2^n}= 1 - 0 = 1$

Yes, I understand all of this. But it continues to beg the question, "what exactly is filling the gap"?

If you want to do the limit for real, look up epsilon-delta proofs. But the point is that we never took a value for N to find this. We simply noted that the bigger N is, the closer $(1/2)^N$ gets to 0. This is not the sort of process where you can imagine that you are summing an very large number of terms. We sum them for an arbitrary N and then we take a limit, which does the job for us.

I studied the epsilon-delta definition of a limit in university. That part of this issue is very clear to me.

 

[Steven G]

But then I am still confused about how the segments/terms reach the other side. A limit just gets arbitrarily close doesn't it? In other words, each n only gets the sum close. How does the gap get filled?

IT DOESN"T get filled. If something get arbitrarily close to a number then it never reaches it.
Suppose you want to get a 4.0 gpa. So far you have 45 A's and just one B. Right now your gpa is 3.97826086957.... Can you ever obtain a 4.0 gpa? What if you only get A's from now on? How many more A's do you need to get that 4.0 gpa? Assume all classes in your college are all 4 credits.

 

[Mates]

I am trying to avoid any miscommunication. So, you agree if I wrote that a partial sum equalled 1? I'll repeat, no partial sum equals 1. A partial sum is just a total of terms 1 through N. Post 21 clearly shows that it can't equal 1 for any N.

No, I said the exact opposite. I was saying that I agree with you that no partial sum can equal 1.

The infinite series sum is another matter. We can't actually total it term by term. But we _define_ its sum as the limit of the partial sum sequence (if it exists). It's not an actual sum. The gap never gets filled. How about this rule - every time you want to say "sum", if it refers to the sum of an infinite series you have to say "so-called sum". This may help you realize that not all sums are created equal.

Ok, let's call it the so-called sum or sum* for short. I can find some proofs if you want that the sum* is 1. Do you want to see them?

 

[Mates]

IT DOESN"T get filled. If something get arbitrarily close to a number then it never reaches it.
Suppose you want to get a 4.0 gpa. So far you have 45 A's and just one B. Right now your gpa is 3.97826086957.... Can you ever obtain a 4.0 gpa? What if you only get A's from now on? How many more A's do you need to get that 4.0 gpa? Assume all classes in your college are all 4 credits.

There are proofs showing that the sum is actually 1. I will find them.

 

[topsquark]

Then if it is not n that fills the gap, how does the gap get filled? There has to be something that is filling the gap. Please just tell me in simple terms exactly what is filling this gap.

The limit procedure "fills the gap." You say you know epsilon-delta proofs. Good! Use them.

I really do not understand what you are not understanding here.

-Dan

 

[Steven G]

IT DOESN"T get filled. If something get arbitrarily close to a number then it never reaches it.
Suppose you want to get a 4.0 gpa. So far you have 45 A's and just one B. Right now your gpa is 3.97826086957.... Can you ever obtain a 4.0 gpa? What if you only get A's from now on? How many more A's do you need to get that 4.0 gpa? Assume all classes in your college are all 4 credits.

There are proofs showing that the sum is actually 1. I will find them.

There is no need to find the proofs. I know that there are INFINITE sums that CONVERGE to 1.

Suppose the sum of 1/2 + 1/4 + 1/8 +... eventually equals 1. Don't you see a problem with that? Because if the sum is eventually equals 1, what happens when you add on the rest of the POSITIVE terms? What other terms you ask? Well, if there are an infinite number of terms, then there must be more terms. If this is true, then the infinite sum will be more than 1. This is what is called a contradiction!!

 

[Mates]

The limit procedure "fills the gap." You say you know epsilon-delta proofs. Good! Use them.

How so? The epsilon-delta just shows that there must always be a gap for any N.

I really do not understand what you are not understanding here.

-Dan

My issue is that I there seems to be two answers to whether or not there is a gap to in infinite sum of 1/2^n. I always thought there would be a gap - I would have argued that until I was blue in the face. But recently I have found a lot information and proofs that suggests that there is no gap.

 

[Steven G]

My issue is that I there seems to be two answers to whether or not there is a gap to in infinite sum of 1/2^n. I always thought there would be a gap - I would have argued that until I was blue in the face. But recently I have found a lot information and proofs that suggests that there is no gap.

Either the proofs that you are talking about are not valid or you are not understanding them.

 

[Steven G]

How so? The epsilon-delta just shows that there must always be a gap for any N.

Yes!
Don't you realize that epsilon-delta is used for limits?!

 

[Mates]

There is no need to find the proofs. I know that there are INFINITE sums that CONVERGE to 1.

Suppose the sum of 1/2 + 1/4 + 1/8 +... eventually equals 1. Don't you see a problem with that? Because if the sum is eventually equals 1, what happens when you add on the rest of the POSITIVE terms? What other terms you ask? Well, if there are an infinite number of terms, then there must be more terms. If this is true, then the infinite sum will be more than 1. This is what is called a contradiction!!

You think you know, but you don't. From Wiki, "There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2."

 

[Steven G]

You think you know, but you don't. From Wiki, "There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2."

I will get feedback from my boss for this, but you're an idiot.

 

[Mates]

Yes!
Don't you realize that epsilon-delta is used for limits?!

But how does the epsilon-delta procedure help "fill the gap" as Topsquark says? I said it does the opposite. It insures there is always a gap for any chosen epsilon.

 

[Mates]

I will get feedback from my boss for this, but you're an idiot.

Okay, I shouldn't have said what I said. I am just tired of not knowing who or what to believe.

 

[Steven G]

Okay, I shouldn't have said what I said. I am just tired of not knowing who or what to believe.

I always believe people who are live, not just someone who wrote something somewhere.
There are some extremely qualified people on this forum. Some have been mathematicians for decades.

 

[lev888]

Okay, I shouldn't have said what I said. I am just tired of not knowing who or what to believe.

Please post links to 2 sources you believe contradict each other.

 

[topsquark]

But how does the epsilon-delta procedure help "fill the gap" as Topsquark says? I said it does the opposite. It insures there is always a gap for any chosen epsilon.

The epsilon-delta method is the formal way of showing that a function approaches a certain value as the argument approaches a certain value.

Get your head off the "gap" thing. You are obsessed over it and it's clouding your thinking. There is only a "gap" if you take your sum to a finite value of N. If you take the partial sum for N you will never get to 1. But if you take the limit as N goes to infinity, the "gap" goes away. The epsilon-delta method will prove that. Again, infinity is not a number and you seem to still be trying to apply it as one.

-Dan

 

[lev888]

The epsilon-delta method is the formal way of showing that a function approaches a certain value as the argument approaches a certain value.

Get your head off the "gap" thing. You are obsessed over it and it's clouding your thinking. There is only a "gap" if you take your sum to a finite value of N. If you take the partial sum for N you will never get to 1. But if you take the limit as N goes to infinity, the "gap" goes away. The epsilon-delta method will prove that. Again, infinity is not a number and you seem to still be trying to apply it as one.

-Dan

Would it be correct to say that any specific positive gap goes away (since we can always find a large enough N that produces a smaller gap), but there always be a gap, it just gets smaller.
Here's a good example. A sum of rational numbers is always rational. The sum of the infinite series in this post is irrational: https://qr.ae/pyLMbI
Can we say that there is no gap?