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

[Mates]

In my last thread a few days ago, it seemed decided that the infinite sum of 1/2^n = 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. Since we got there presumably using only n from the set of natural numbers, wouldn't this mean that the final n would have to equal infinity? If not, what else could it be?

In other words, wouldn't it have to be the term 1/infinity, or the "infinity-ith" point?

If I am accurate, how can it be said that the natural numbers can numerate all points of a set of size aleph-null?

 

[topsquark]

In my last thread a few days ago, it seemed decided that the infinite sum of 1/2^n = 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. Since we got there presumably using only n from the set of natural numbers, wouldn't this mean that the final n would have to equal infinity? If not, what else could it be?

In other words, wouldn't it have to be the term 1/infinity, or the "infinity-ith" point?

If I am accurate, how can it be said that the natural numbers can numerate all points of a set of size aleph-null?

First, a correction: infinity is not a number. It has no size, no value. There is no final end in that series.

The infinite sum may be expressed as
[imath]\displaystyle \lim_{N \to \infty} \sum_{n=1}^{N} \dfrac{1}{2^n}[/imath]

The concept is that the summation gets closer and closer to a specific value as n grows without bound, which was explained in the previous thread, as I recall.

Now, as to enumeration, we can show that there is always a natural number bigger than any other natural number, so they have no maximum. We define the cardinal number (the measure of the size of a set) [imath]\aleph_0[/imath] to be the "number of elements of" or size of the natural numbers. Now, [imath]\aleph_0[/imath] has several other properties that might seem to be unrelated to this, but this is the actual definition of it.

-Dan

 

[Steven G]

Consider the following. 1 + 2 + 3 + 4 + 5 = 15. Does 15 start at 1? Does 15 end at 5? What does that even mean?!

There are many infinite series that sum to 1. Just find any infinite series whose sum is finite. Call this sum k. Now consider a new infinite series where we divide each term of that infinite series (the one that sums to k) by k. Then the sum will be 1. The first term will not be 1/2. So infinity does not start at 1/2!

 

[lev888]

beginning point at 1/2 and a definite end point at 1

1/2 and 1 represent a partial sum and the limit of the partial sum sequence respectively. Infinity refers to the number of terms in the sequence. These are totally different concepts. Not sure in what sense these 2 values are the start and end points of infinity.

 

[Mates]

First, a correction: infinity is not a number. It has no size, no value. There is no final end in that series.

The infinite sum may be expressed as
[imath]\displaystyle \lim_{N \to \infty} \sum_{n=1}^{N} \dfrac{1}{2^n}[/imath]

The concept is that the summation gets closer and closer to a specific value as n grows without bound, which was explained in the previous thread, as I recall.

Now, as to enumeration, we can show that there is always a natural number bigger than any other natural number, so they have no maximum. We define the cardinal number (the measure of the size of a set) [imath]\aleph_0[/imath] to be the "number of elements of" or size of the natural numbers. Now, [imath]\aleph_0[/imath] has several other properties that might seem to be unrelated to this, but this is the actual definition of it.

-Dan

For n to complete the summation from 0 to 1, doesn't it have to have equaled more than any natural number?

 

[lev888]

For n to complete the summation from 0 to 1, doesn't it have to have equaled more than any natural number?

It seems you don't quite understand what a limit of a sequence is. The partial sums get closer and closer to 1, but no member of the sequence ever reaches 1, as we discussed in the other thread. There is no completion of the summation. It equals 1 only in the limit.

 

[Mates]

Consider the following. 1 + 2 + 3 + 4 + 5 = 15. Does 15 start at 1? Does 15 end at 5? What does that even mean?!

If you don't try to understand what I am saying, we will go down an infinite rabbit hole of unnecessary semantic "misunderstandings".

I meant the summation of infinity.

 

[Mates]

1/2 and 1 represent a partial sum and the limit of the partial sum sequence respectively. Infinity refers to the number of terms in the sequence. These are totally different concepts. Not sure in what sense these 2 values are the start and end points of infinity.

Like I told the other poster, I meant the first term and the last term of the summation of infinity as being the ends.

Just a side rant: (It seems that it is somehow ingrained into our human nature that when we read something on a forum that is clearly a mistake (due to the context), we don't try to understand what that poster might have meant. Instead we choose a ridiculous possibility that makes absolutely no sense given the context thus giving the writer 0 credit in making any sense in the post.)

 

[Mates]

It seems you don't quite understand what a limit of a sequence is. The partial sums get closer and closer to 1, but no member of the sequence ever reaches 1, as we discussed in the other thread. There is no completion of the summation. It equals 1 only in the limit.

I thought that we concluded that there is a completion. That is why I started this thread.

 

[lev888]

Like I told the other poster, I meant the first term and the last term of the summation of infinity as being the ends.

Just a side rant: (It seems that it is somehow ingrained into our human nature that when we read something on a forum that is clearly a mistake (due to the context), we don't try to understand what that poster might have meant. Instead we choose a ridiculous possibility that makes absolutely no sense given the context thus giving the writer 0 credit in making any sense in the post.)

Nobody is getting graded here. Math requires using precise terms. If you made a mistake just correct it in the next post. It's difficult to make assumptions about your real intent. For example, even with your last clarification I _still_ don't understand what you mean. As has been stated many times, there is no last term of the summation!

 

[Mates]

Nobody is getting graded here. Math requires using precise terms. If you made a mistake just correct it in the next post. It's difficult to make assumptions about your real intent. For example, even with your last clarification I _still_ don't understand what you mean. As has been stated many times, there is no last term of the summation!

The reason why there seems to be a final term is because there is no gap between the 1/2 and 1. Just like with the bridge analogy yesterday, there is no gap left, right?

 

[topsquark]

For n to complete the summation from 0 to 1, doesn't it have to have equaled more than any natural number?

That's what [imath]\displaystyle \lim_{N \to \infty}[/imath] does. Just to be clear,
[imath]\displaystyle \lim_{N \to \infty} f(N)[/imath]

means we are going to take larger and larger values of N and see what happens to f(N). If f(N) approaches some finite value, C, then we write
[imath]\displaystyle \lim_{N \to \infty} f(N) = C[/imath]

But note that we need to prove that f(N) approaches this constant. We are not actually going to calculate any values f(N) when we do this proof.

-Dan

 

[lev888]

I thought that we concluded that there is a completion. That is why I started this thread.

The sum is finite since the sequence of partial sums has a limit. But the sequence is infinite. There is no last term.

 

[Steven G]

Just like with the bridge analogy yesterday, there is no gap left, right?

I can't imagine that I am wrong thinking that you really mean that there is no gap left. I don't accept that. There is NO n such that 1/2ⁿ=0. The problem is that 1/2ⁿ=0 is necessary for there to be no gap.

Are you a college student? What is your major? I ask as my posts might be different after hearing your answers.

 

[Mates]

That's what [imath]\displaystyle \lim_{N \to \infty}[/imath] does. Just to be clear,
[imath]\displaystyle \lim_{N \to \infty} f(N)[/imath]

means we are going to take larger and larger values of N and see what happens to f(N). If f(N) approaches some finite value, C, then we write
[imath]\displaystyle \lim_{N \to \infty} f(N) = C[/imath]

But note that we need to prove that f(N) approaches this constant. We are not actually going to calculate any values f(N) when we do this proof.

-Dan

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?

 

[Mates]

There is no last term.

I know that the sequence is infinite, yet the gap still gets filled. What fills the gap?

 

[Mates]

I can't imagine that I am wrong thinking that you really mean that there is no gap left. I don't accept that. There is NO n such that 1/2ⁿ=0. The problem is that 1/2ⁿ=0 is necessary for there to be no gap.

You are right. I thought we all concluded on the other thread that the gap does get filled.

Are you a college student? What is your major? I ask as my posts might be different after hearing your answers.

No, not anymore. I took first-year calculus though.

 

[lev888]

I know that the sequence is infinite, yet the gap still gets filled. What fills the gap?

The gap never gets filled for any finite number n. It "gets filled" at the limit - i.e. if it WERE possible to plug in infinity for n, what would be the sum of the series? It would be 1. We know that it is not possible. But we still can talk about the sum to which the series converges.
Maybe you are more comfortable with functions? Consider y=1/x. The limit of the function as x goes to +infinity is 0. It gets closer and closer to 0, but NEVER gets there. The limit is a useful concept that lets us talk about various properties of such functions and points.

 

[Mates]

The gap never gets filled for any finite number n. It "gets filled" at the limit - i.e. if it WERE possible to plug in infinity for n, what would be the sum of the series? It would be 1.

But it is 1. That's what the other thread was all about.

 

[lev888]

But it is 1. That's what the other thread was all about.

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