Undergrad Ambiguities of the "...." notation

[Stephen Tashi]

A popular response to a novice asking whether .999... = 1 is for an expert to demand that novice define what the notation ".999..." means. I think this is the best response. However, I note that many experts tend to accept the "..." notation in other contexts without demanding an explanation.

The video points out ambiguities in interpreting the "..." notation.

He points out that Ramanujan's famous proof that
$\sqrt{ 1 + 2 {\sqrt{1 + 3 \sqrt{1 + 4 ...}}}} = 3$ is invalid.

At 8;15 he points out two different plausible interpretations of infinitely continued fractions.

 

[Drakkith]

Very nice video. I've seen several Mathologer videos before and they tend to be very enjoyable and insightful, with excellent (and accurate) explanations about things that are often confusing or used incorrectly.

 

[PeroK]

The three dots "..." are formally called "ellipsis" and to use them is being "elliptical", which can also mean omitting details so as to be hard to understand.

In these cases, therefore, the presenter could be elliptical, in both senses of the word.

 

[SSequence]

I have rather what is a dumb (related) question (probably related to analysis). Consider a set of real (or rational) numbers placed on $\mathbb{N}^2$ grid. Call the n-th horizontal line $h_n$ and the n-th vertical line $v_n$.
$h_n:=(0,n),(1,n),(2,n),(3,n),(4,n)...$
$v_n:=(n,0),(n,1),(n,2),(n,3),(n,4)...$

Does it matter if:
(1) we add the numbers in $h_0$. If they converge, then call it $r_0$. Then we add all the numbers in $h_1$. If they converge, then call it $r_1$. If all the horizontal lines converge, then we make the sum: $r_0+r_1+r_2,...$. If this converges then we call this number $H$.

(2) Same as (1) except we along vertical lines. If everything turns out convergent we call the resulting number $V$.

(3) We make a sum of all the numbers using some elementary encoding function. If the sum turns out convergent, we call it $E$.

How are H,V and E are related generally (if they are)?

 

[Mark44]

I have rather what is a dumb (related) question (probably related to analysis). Consider a set of real (or rational) numbers placed on $\mathbb{N}^2$ grid. Call the n-th horizontal line $h_n$ and the n-th vertical line $v_n$.
$h_n:=(0,n),(1,n),(2,n),(3,n),(4,n)...$
$v_n:=(n,0),(n,1),(n,2),(n,3),(n,4)...$

It's unclear to me what you're trying to say here. Do the expressions such as (3, n) represent the rational number $\frac 3 n$? If these are instead real numbers, how does an expression such as (3, n) map to a real number.

Supposing these are rational numbers, does adding, say, row 2, $\frac 0 2 + \frac 1 2 + \frac 2 2 + \dots + \frac 4 2 + \dots$ have even a chance of converging?
Going down a column, you have what is essentially a harmonic series, which is known to diverge.

Does it matter if:
(1) we add the numbers in $h_0$. If they converge, then call it $r_0$. Then we add all the numbers in $h_1$. If they converge, then call it $r_1$. If all the horizontal lines converge, then we make the sum: $r_0+r_1+r_2,...$. If this converges then we call this number $H$.

(2) Same as (1) except we along vertical lines. If everything turns out convergent we call the resulting number $V$.

(3) We make a sum of all the numbers using some elementary encoding function. If the sum turns out convergent, we call it $E$.

How are H,V and E are related generally (if they are)?

 

[SSequence]

It's unclear to me what you're trying to say here. Do the expressions such as (3, n) represent the rational number $\frac 3 n$? If these are instead real numbers, how does an expression such as (3, n) map to a real number.

Sorry I wasn't precise enough. This isn't what I intended. What I imagined was "arbitrary" real/rational numbers placed on each position corresponding to ω².

So, for example, denote the (unique) real number that is placed at each position corresponding to some α<ω² as R(α).

So this is what (1) in post#4 will translate to:
Consider the sum $R(0)+R(1)+R(2)...$. Is it convergent or not? If it is then call it $r_0$.
Now consider the sum $R(\omega)+R(\omega+1)+R(\omega+2)...$. Is it convergent or not? If it is then call it $r_1$.
Now consider the sum $R(\omega \cdot 2)+R(\omega \cdot 2+1)+R(\omega \cdot 2+2)...$. Is it convergent or not? If it is then call it $r_2$.
and so on...

If all the individual $r_i$'s ($i \in \mathbb{N}$) are convergent then consider the sum:
$r_0+r_1+r_2+r_3+...$
If this is convergent too then call it $H$.

=========================

For example, now we can ask the relation between the following propositions:
p: $H$ is convergent
q: $V$ is convergent
r: $E$ is convergent

And ask about truth value of:
If $H$ and $V$ are both convergent then we always have $H=V$

and so on.

 

[mfb]

You need absolute convergence, otherwise it doesn't work and you can get different results.

 

[SSequence]

Any specific and easily described example where H and V both exist but are different from each other?

 

[mfb]

I don't have a nice example now. It is easy to construct examples where e.g. the row sums diverge but the column sums do not, of course.

 

[SSequence]

Actually, that would also be of some independent interest separately I think (in addition to the one mentioned in post#8 and perhaps others**).

** By others I mean the combination of possibilities such as ones that could occur in post#6

 

[Stephen Tashi]

Some notation in search of a definition:

$\begin{matrix} 1&+&(-1/2)&+&{1/3}&+&(-1/4)&+& (1/5)&+&{ ...} \\ + & \ & + & \ &+ & \ &+ &\ & + & \ & {...} \ \\ (-1/2)&+&1/3 &+&(-1/4)&+& (1/5)&+&(-1/6) &+ &{ ...}\\ + & \ & + & \ &+ & \ &+ &\ &+ &\ &+ & {...} \\ 1/3 &+&(-1/4)&+& (1/5)&+&{ ...} & \ & {...} & \ & {...}\\ + & \ & + & \ &+ & \ &+ &\ & {...} \ \\ {...}& \ & {...} & \ &{...} & \ &{...} & \ & {...} \end{matrix}$

 

[Stephen Tashi]

A homework problem somewhat similar to @SSequence 's questions: https://www.physicsforums.com/threads/cesaro-limits.955003/

No helpers have appeared yet.

 

[ad infinitum]

A popular response to a novice asking whether .999... = 1 is for an expert to demand that novice define what the notation ".999..." means. I think this is the best response. However, I note that many experts tend to accept the "..." notation in other contexts without demanding an explanation.

The video points out ambiguities in interpreting the "..." notation.

He points out that Ramanujan's famous proof that
$\sqrt{ 1 + 2 {\sqrt{1 + 3 \sqrt{1 + 4 ...}}}} = 3$ is invalid.

At 8;15 he points out two different plausible interpretations of infinitely continued fractions.

It is not that expert accept the "..." notation whithout explanation but rather that the experts are writing there work for either other experts who already know the explantion at assume it implicitlyan already consider it to be common knowledge or for student who are learning the material in order to become experts and need to become familiar with the notation of the field. The reason we ask for clarification for the ".999..." notation is that newcomers must learn to understand this distinction.