Like (\infty)?
I am breaking the pattern. That was my point.
But you want to put (\infty) in there. So let’s talk about (\infty) as a fuzzy blob of impossibly big numbers, not as limitlessness, as the spectrum of numbers you would find if you transcended the limits of the limitless.
2.5 is not a natural number so this makes no sense.
Yes.
Why would you break the pattern? (Assuming you know what I mean.) Obviously, if you break the pattern, you might not get a number that is less than (1).
In the case of (\sum_{i=1}^{n}), (i) starts with (1), increases by (1) and ends with (n). The number of terms is (n), so the sum stops (is complete) after (n) number of terms.
In the case of (\sum_{i=1}^{\infty}), (i) starts with (1), increases by (1) and does not end. The fact that (i) does not end tells us that the number of terms is (\infty). This means the sum stops (is complete) after an infinite number of terms. (I assume you’re one of the people in this thread who have no problem with the concept of “actual or completed infinity”.)
(\sum_{i=1}^{\infty}) does not mean that (i) ends with (\infty). In other words, (\sum_{i=1}^{\infty} i) is not equal to something like (1 + 2 + 3 + \cdots + \infty). (Note that such a sum would have more than (\infty) terms.)
The value of (i) is always a natural number. So if we are asking a question such as “What’s the value of the sum of terms whose index is (i, 1 \leq i \leq x)?” then (x) cannot be anything other than a natural number since the range of (x), in such a case, must be the range of (i) – and this means that (x) can’t be (\infty). But if we’re asking a question such as “What’s the value of the sum after (x) number of terms?” then the range of (x) goes from (1) to the number of terms of the sum. If the number of terms is infinite, then (x) can be (\infty). And it is precisely this question that we’re asking.
What is the value of the sum (\sum_{i=1}^{\infty} \frac{9}{10^i}) after an infinite number of terms?
You’re saying it’s (1), I am saying it’s less than (1).
My argument (which is basically James’s argument) is that the pattern of this sum prohibits its value after (x) number of terms to be equal to (1) for any (x > 0). (You can limit the value of (x) to numbers that have no fractional component, if you want.) Since (\infty) is greater than (0), it applies to (\infty) as well.
By what logic does the value of this sum become (1) after an infinite number of terms?
(\sum_{i=1}^{n} \frac{9}{10^i} = 1 - \frac{1}{10^n})
In order for (\sum_{i=1}^{n} \frac{9}{10^i}) to be equal to (1), there must be (n) such that (1 - \frac{1}{10^n} = 1).
This means that there must be (n) such that (\frac{1}{10^n} = 0).
Suppose that (x) is such a number: (1 \div 10^x = 0).
This means that (0 \times 10^x = 1).
But zero times any number is zero, right?
Magnus! Honestly! sigh
This is what I was trying to explain to you before.
If you halve 1, it’s 1/2 + 1/2
If you halve THAT!
It’s 1/4+1/4+1/4+1/4. Etc…
If you keeps doing this:
At infinity (convergence)
EVERY!!! Whole number solves as zero.
I still think you are confusing all your arguments. You switch back and forth when it suits you without understanding (or caring) about the implications.
Well, you didn’t say much. What you said is basically:
(1 = \frac{1}{2} + \frac{1}{2} = 2 \times \frac{1}{2})
(1 = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 4 \times \frac{1}{4})
(\dotso)
(1 = 0 + 0 + 0 + \cdots = \infty \times 0)
How can that be the case when zero times anything is zero?
Properly speaking, (1 = \frac{1}{\infty} + \frac{1}{\infty} + \frac{1}{\infty} + \cdots = \infty \times \frac{1}{\infty}).
I said a lot! I said that it’s impossible for convergence theory to be true (standard calculus and other mathematics)
Ultimately it equals (according to convergence theory)
0+0+0+0 (etc…) at infinity! (Convergence)
That means that every whole number equals zero. (Contradiction)
How’s that relevant to anything of what I’m saying?
How’s what you said relevant to what I said?!?!
1/infinity is a meaningless mathematical phrase.
You quoted me. I guess it’s somehow relevant to what I said in that quote.
(\frac{1}{\infty}) is a number greater than (0) but smaller than every number of the form (\frac{1}{n}, n \in N). That’s what it means (so it’s not really meaningless.)
(0 < \frac{1}{\infty} < \cdots < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{1}{1})
1/infinity is not a number. It’s hypothetically, a procedure. It is most definitely not a number.
It’s not a procedure.
(\infty) represents a number greater than every number of the form (n, n \in N).
Similarly, (\frac{1}{\infty}) represents a number greater than (0) but less than every number of the form (\frac{1}{n}, n \in N).
(0 < \frac{1}{\infty} < \cdots < \frac{1}{3} < \frac{1}{2} < \frac{1}{1} < 2 < 3 < \cdots < \infty)
Also:
(0.\dot9 + 0.\dot01 = 1)
Note that (0.\dot01) or (\frac{1}{10^\infty}) is actually smaller than (\frac{1}{\infty}).
Ok, fine. I don’t buy this, but let’s say that what you’re saying is absolutely true.
Per the argument I leveled. That means every whole number is EXACTLY equal to the lowest possible “number” (your argument, not mine) that’s not zero.
My argument still stands. It’s absurd.
How did you arrive at the conclusion that every whole number is exactly equal to the lowest number?
That’s not even true for (\frac{1}{\infty}) let alone for whole numbers.
Really Magnus ?!
I’ll use your own post for it!
viewtopic.php?p=2758485#p2758485
You’re right. 1=0 is a constradiction.
My argument proves that when numbers converge at infinity (and in saying this, infinity is NOT A NUMBER!)
That 1=0.
Thus, infinities do not converge.
All you did was change infinity to “lowest possible ‘number’ that’s not equal to zero, which by my argument, makes every whole number equal to “the lowest possible number not equal to zero” which is still a contradiction.
That means that 1=2!! Contradiction !
What argument, Ecmandu? Where is it?
You replied to the post yourself ! Honestly! This is getting absurd!
What exactly does that prove?
Just what I said it does. Infinite series don’t converge.
To answer your question.
This started out with you asking:
“You’d have to explain why you’re limiting yourself to integers.”
To which I said: “Try it without integers. It doesn’t work.”
I was showing you it doesn’t work. Of course, what you really meant was the non-integer (\infty).
I don’t get the distinction between these two cases. Sounds like the exact same case just worded differently. In the one case you’re saying x can equal (\infty), in the other that it can’t.
This logic:
(x = 0.\dot9)
(10x = 9.\dot9)
(10x = 9 + 0.\dot9)
(10x = 9 + x)
(9x = 9)
(x = 1)