My “copy and paste” examples are dealing with completed infinity. There is a step for every n ∈ ℕ. Infinitely many steps. All finite numbers are there and all very very large finite numbers are there. I am not stopping at a specific very very large finite number.
No you don’t. As you have said yourself, there is no infinite step, no “completion”.
It doesn’t take an infinite step to complete infinity. It simply takes infinitely many steps to complete infinity. One step for each n ∈ ℕ.
What happens at infinity then? And remember “undefined” is not a valid answer.
No infintieth step - no action at an infintieth step - nothing happens then.
By “happens” I meant “What is the value of the expression at infinity”. If you, as you write, have a “completed infinity” what is the value of it? Undefined is not an acceptable answer, mathematically.
You know that the value at infinity is defined by the actions of the infinitely many steps. Why don’t you tell me, what the value at infinity could be then?
Here’s another interesting exercise: We agree, that the limit of 0.5 + 0.25 + 0.125 + … is 1.
What is the limit, if we add a small number at each step, like (0.5+0.015625) + (0.25+0.015625) + (0.125+0.015625) + …
Now choose an even smaller number of the set { 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, … } instead of 0.015625. What is the limit?
We’re not “completing” infinity … we’re approaching infinity … and you’ve already agreed that as n -> ∞, then (0.999…) -> 1 … and for any n ∈ ℕ, n ≠ ∞ … however we can still say (0.999…) = 1 … by the fundamental theories of calculus.
Maybe your Algebra teacher will loan you their calculus textbook …
In the first case, the limit is 1.015625 … in the second the limit is 1 + a, where a is from the set specified …
1
It’s very strange that you haven’t realized by now that everyone but you consider it to be 1.
You say we have a “completed infinity”. What’s its value?
You missed that he’s adding the number to every step. Which means the limit is infinity.
To netzwelter: How is this an interesting exercise? As long as you, as you did, choose a constant number, any constant number, then the series diverges. If, as you’re doing in this particular “exercise” pick the n-th term of a converging series S and add it as a constant to every term in that series, by the n+1th term it’s the dominant part of every term and the partial sums of the first k terms converges towards k*S[sub]n[/sub]. Let k approach infinity and the new series diverges.
Thank you for this correction … I’ll re-double my efforts to insure a full belly of coffee before I post in the morning …
The precise value? It’s 9(⅒) + 9(⅒)² + 9(⅒)³ + …
What do you suggest? That there is no number a in the set { 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, … } small enough to have (0.5+a) + (0.25+a) + (0.125+a) + … resulting in less than infinity? Although the addition of infinitely many of the numbers of the set result in a limit of 1?
The first is a convergent series; and has the limit of 1 … the second is a divergent series, and has the limit of infinity …
In the first case, the value of the series can equal it’s limit, therefore it does … in the second case the value of the series cannot equal it’s limit, so here we can say the value is “undefined” …
This may seem a journey into semantics … but you present two different cases and each case is evaluated differently … here, we’re focused on the first case, where the series can equal it’s limit …
Why do you think it shouldn’t?
Eh, yes. If you grab an a from the set you might as well ignore the rest of your definition. Pick a large enough n and you can approximate the sum as a*n where n is the number of terms you sum over.
So unless a = 0, you will have a divergent series. Do you really not see that this is trivial and irrelevant?
I want to assure that we are dealing with the same notion. So, we agree that every number of the set { 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, … } is far from being infinitesimally small.
Now let’s take a look at this:
(1) Do you also agree that we can write
0.9
0.99
0.999
…
as
[0, 0.9]
[0, 0.9]∪[0.9, 0,99]
[0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]
…?
(2) Do you also agree that
0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…?
(3) Is there a segment in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… which is not already present in the list
[0, 0.9]
[0, 0.9]∪[0.9, 0,99]
[0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]
…?
No …
Why do you think the value of our previous function cannot equal it’s limit?
I don’t want to cut you off but it doesn’t matter how you express it, it’s still equal it one.
You’re not speaking to me but I am mighty bored with the dull repetition of your dull arguments.
(1) No, 0.9 is a real number, [0, 0.9] is a segment of the real number line.
(2) No, see (1)
(3) Yes, [0, 1]
Please answer the 3 questions and you will see what I mean.
So, what’s the difference?