Cont: Deeper than primes - Continuation 2

[jsfisher]

Very simple:

Since no term of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) can be used in order to conclude anything about the sum of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) = 0.999...₁₀

You have started your "proof" with a false statement.

...(and in this case 0 < |a_n - L| < ε)

You also need to make clear what you are referring to. This entire parenthetical remark has no established relationship to what preceded it.

...then a_n can't be used

Can't be used for what?

...so the sum over a_|N| = 0.999...₁₀ is used

Since a_|N| does not exist, your statement is nonsense.

...and L=1, so (0 < |0.999...₁₀ - 1|) AND (0.999...₁₀ = 1) is false.

Yes, 0 < | 0.999... - 1| would be a false statement. What does that have to do with anything?

Let X be a placeholder for that is summed.

If |X - L| = 0 then L is called the limit of X.

Without a method to evaluate the summation over an infinite sequence, your attempt to define 'doron-limit' is a failure.

So, let's recap. Doronshadmi demonstrates the fallacy of both limits and series using the technique of proof by strawman. He then attempts to define his version of limit by not defining it at all.

 

[doronshadmi]

You have started your "proof" with a false statement.

Wrong, your "proof" uses finite numbers in order to conclude something about number 0.999...₁₀

Since a_|N| does not exist, your statement is nonsense.

(0.9₁₀, 0.09₁₀, 0.009₁₀ ) is a concrete example of a_|N|, so you have no case.

 

[doronshadmi]

Let's clarify the difference between a_n and a_|N|, by using (0.9₁₀, 0.99₁₀, 0.999₁₀,... ) or (0.9₁₀, 0.09₁₀, 0.009₁₀,... ), without loss of generality.

By the standard definition where 0 < |a_n - L|, a_n is some term of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,... ) that is arbitrary closer to L, so your method clearly uses the finite in order to conclude something about the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀,... ) |N| infinite terms , which is a fundamental fallacy of your method.

On the contrary, by my method a_|N| is the sum over sequence (0.9₁₀, 0.09₁₀, 0.009₁₀,... ) such that |a_|N| - L| > 0 by 0.000...1₁₀, where the 0.000... part is a place value keeper that is in bijection with all the place values of 0.999...₁₀, where in both cases there are no more than |N| places.

As a result |0.999...₁₀ - 1| > 0 by the relative difference between |N| and |R|, such that the 0.000... part of number 0.000...1₁₀ is at |N| level and the ...1₁₀ part of number 0.000...1₁₀ is at |R| level, and therefore it is taken as a one element > 0 (from the point of view of |N| level) that closes the gap between 0.999...₁₀ and 1.

 

[jsfisher]

You have started your "proof" with a false statement.

Wrong, your "proof" uses finite numbers in order to conclude something about number 0.999...₁₀

You cannot make your false statement true by attacking mine. That form of fallacious argument even has a name. So you have done nothing to defend your bogus statement.

On the other hand, your attack on my statement is also mired in a logical fallacy. The meaning of an infinite series is a matter of definition. Try as you may, you cannot disprove definitions.

Since a_|N| does not exist, your statement is nonsense.

(0.9₁₀, 0.09₁₀, 0.009₁₀ ) is a concrete example of a_|N|, so you have no case.

Really? Well, let's see. I guess we can assume you meant for (0.9, 0.09, 0.009) to be a sequence known as A. We will assume also that a_i is the i-th element of the sequence and that the index has its origin at one.

I think we can all agree with the following:

a₁ = 0.9
a₂ = 0.09
a₃ = 0.009

...but that is it. A sequence of three elements is unlikely to have more than three elements. Certainly not one that could be considered the |N|-th element.

Perhaps you left out the ellipses in presenting the sequence. Your attention to detail is well documented in this thread and others, so such a typographic error would surprise no one. A = (0.9, 0.09, 0.009, ...). Still no |N|-th element. Or maybe you can identify the element you meant by a_|N|.

From your variety of fallacious arguments and omission of significant detail we end up with...nothing.

 

[jsfisher]

Let's clarify the difference between a_n and a_|N|, by using (0.9₁₀, 0.99₁₀, 0.999₁₀,... ) or (0.9₁₀, 0.09₁₀, 0.009₁₀,... ), without loss of generality.

Clarification should be unnecessary if you were not abandoning Mathematics in favor of your own undefined terminology and notation.

By the standard definition where 0 < |a_n - L|

Full stop. Standard definition of what? What does L represent? What do a and n represent? Details are important, Doronshadmi. What is it you think your are talking about here?

 

[doronshadmi]

so such a typographic error would surprise no one.

A waste of energy of yours on some typographical error of mine.

Or maybe you can identify the element you meant by a_|N|.

a_|N| is the sum over sequence (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) and not one of its terms, unlike in the case of a_n which is some finite number (some term) of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that is arbitrarily closer to 1.

Since your notions are fixed only on a_n case, you simply unable to understand a_|N| (the difference between them is very simply explained and demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=10643978&postcount=463, but as usual it goes above your head exactly because you get a_|N| in terms of a_n).

 

[doronshadmi]

Clarification should be unnecessary if you were not abandoning Mathematics in favor of your own undefined terminology and notation.

My terminology is defined. It simply not your false terminology that forces the finite in order to conclude something about the infinite.

Full stop. Standard definition of what?

Don't be so dramatic. The standard definition of limit, which forces the finite in order to conclude something about the infinite.

What does L represent? What do a and n represent? Details are important, Doronshadmi. What is it you think your are talking about here?

This time please read all of http://www.internationalskeptics.com/forums/showpost.php?p=10643978&postcount=463 before you reply.

 

[doronshadmi]

It was a failure when you first posted that.

Wrong, it is perfectly valid.

Please demonstrate your method using the sequence {1, -1/2, 1/3, -1/4, 1/5, ...}

{1, -1/2, 1/3, -1/4, 1/5, ...} is not a sequence (and I not going to waste my energy on your typos, as you did about mine in http://www.internationalskeptics.com/forums/showpost.php?p=10644301&postcount=464).

Any way, your one step problem is already addressed in http://www.internationalskeptics.com/forums/showpost.php?p=10637196&postcount=437 even if you don't get it because of your forcing the finite on the infinite attitude.

 

[jsfisher]

A waste of energy of yours on some typographical error of mine.

If your errors of omission were infrequent, we wouldn't be left with so much guessing about what you really had meant. Perhaps you can do better.

a_|N| is the sum over sequence (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) and not one of its terms, unlike in the case of a_n which is some finite number (some term) of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that is arbitrarily closer to 1.

Another bit of corruption of established notation. Nope. Not acceptable. Besides, we already have several ways of expressing that. The simplest, perhaps, is 0.999.... Another isn't needed, especially one that is so unrelated to what's being expressed.

 

[jsfisher]

Don't be so dramatic. The standard definition of limit, which forces the finite in order to conclude something about the infinite.

No where in the definition of 'limit' does 0 < |a_n - L| occur.

Even it if it (which it doesn't), you'd still need to connect it to your thoughts, here.

 

[jsfisher]

{1, -1/2, 1/3, -1/4, 1/5, ...} is not a sequence

Would that be because you believe the only notation for sequences allowed requires parentheses? If that makes you feel better, we'll go with this:

Please show us how you would apply method to "add in one step" to derive the series for (1, -1/2, 1/3, -1/4, 1/5, ...).

 

[doronshadmi]

Another bit of corruption of established notation.

Your, so called, established notation expresses corrupted notion exactly because a_n forces the finite on the infinite.

 

[doronshadmi]

The simplest, perhaps, is 0.999.... Another isn't needed, especially one that is so unrelated to what's being expressed.

Simply wrong, 0.999...₁₀ is some particular case of the general notation of a_|N| as the sum of any sequence with |N| convergent terms.

 

[doronshadmi]

Please show us how you would apply method to "add in one step" to derive the series for (1, -1/2, 1/3, -1/4, 1/5, ...).

http://www.internationalskeptics.com/forums/showpost.php?p=10637196&postcount=437.

Some hint: a_n can't help you to get it but a_|N| can.

 

[doronshadmi]

No where in the definition of 'limit' does 0 < |a_n - L| occur.

Please demonstrate even one case of 0 = |a_n - L|, such that a_n is some partial sum that is a term of sequence (s₁, s₂, s₃, ...) that is arbitrarily closer to L.

 

[jsfisher]

Simply wrong, 0.999...₁₀ is some particular case of the general notation of a_|N| as the sum of any sequence with |N| convergent terms.

That is not what you said before. Be that as it may, it is still a corruption.

 

[jsfisher]

http://www.internationalskeptics.com/forums/showpost.php?p=10637196&postcount=437.

Some hint: a_n can't help you to get it but a_|N| can.

All you have done so far is repeatedly claimed the "all in one step" summation is possible. You have yet to show how. Show us how it is possible using the sequence (1, -1/2, 1/3, -1/4, 1/5, ...).

 

[jsfisher]

No where in the definition of 'limit' does 0 < |a_n - L| occur.

Please demonstrate even one case of 0 = |a_n - L|, such that a_n is some partial sum that is a term of sequence (s₁, s₂, s₃, ...) that is arbitrarily closer to L.

Doronshadmi,
You are the one that made the bogus claim that "0 < |a_n - L|" was some part of the definition of 'limit'. It is for you to either support that bogus claim or retract it.


Whether or not I can exhibit a sequence of partial sums where in some element of that sequence equals the sequence's limit, which I can, assuming that's what you meant by your ill-presented notation, is not relevant to you supporting your bogus claim regarding the definition of 'limit'.

 

[doronshadmi]

Whether or not I can exhibit a sequence of partial sums where in some element of that sequence equals the sequence's limit, which I can,

So please show such an element.

 

[doronshadmi]

That is not what you said before.

Please show what I said before, that prevent a_|N| of being a general notation of the sum of some convergent series with |N| terms, to some limit.