Math stuff

[Luke]

On 2/5/2019 at 12:50 AM, DavidW said:

If 1/infinite is shorthand for lim_(n->infinity) 1/n, then it is zero.

Why?

As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero?

(I'm assuming you too have a background similar to mine and/or a degree/PhD in Mathematics, which is very likely...)

Edited March 31, 2019 by Luke

[grodrigues]

@Luke:

 

""As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero? "

 

Your question is baffling because you give the answer in it. A limit of a sequence or function  (that it is *the* limit, when it exists, is a theorem) is *by definition* a *real number* (let us stick to real analysis here) that satisfies a certain property, "the value" in your sentence. Period. *The* limit lim_{n->0}(1/n) *is* a real number, namely 0; adding "exactly" does no work and conveys absolutely nothing.

 

\math

[Luke]

@grodrigues

Are you talking about the fact that properties should not be used to give definitions?

[DavidW]

2 hours ago, Luke said:

Why?

As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero?

(I'm assuming you too have a background similar to mine and/or a degree/PhD in Mathematics, which is very likely...)

The definition of the limit of a sequence x_0, x_1, x_2, ...x_n, .... is that number x (if there is one) such that for any e>0, you can find some n so that all the sequence members after x_n are within e of x.

so if our sequence is 1, 1/2, 1/3, 1/4, ..., 1/n, ... then zero satisfies the definition (the sequence gets arbitrarily close to it) so it’s the limit. It’s fairly easy to show that if a series has a limit, it’s unique.

Note: we’re not saying that the sequence eventually gets to 0: any member of the sequence is  non-zero. But zero is still the limit, i.e. the value that sequence elements get arbitrarily close to.

[grodrigues]

@Luke:

 

"Are you talking about the fact that properties should not be used to give definitions? "

 

No. where did you get that idea? What I was saying, or tried to say, was (1) by *definition* the *limit of a sequence*, when it exists, is a *real number* (once again, sticking to real analysis) satisfying a certain property that David gave (in symbols, the sequence x_n converges to x if (by definition) A e > 0 E n_0 \in N A n > n_0: |x_n - x| < e with N the set of natural numbers, A the universal quantifier and E the existential one) and (2) in your very sentence you speak about the limit as " the value that", so unless you are using the words in a total novel way, it is baffling (at least to me) why you are asking "how can it be exactly zero?"

 

\math

[Luke]

18 minutes ago, grodrigues said:

@Luke:

 

"Are you talking about the fact that properties should not be used to give definitions? "

 

No. where did you get that idea?

From the fact that I didn't state the definition of a limit but just a property..... Sorry if I misunderstood your words......

Anyway, it seems both you and @DavidW know a lot about math, I may ask you clarifications about some theorems (maybe ....)

[Luke]

@DavidW, @grodrigues

Do you perhaps know how to make this "box" in LaTex when the document class is article?

EDIT: OK, don't worry, I found them...... I was looking for something like this:

\begin{tcolorbox}[colback=blue!5!white,colframe=blue!75!black,title=My title]
  My box with my title.
\end{tcolorbox}

 

Edited March 31, 2019 by Luke