Calculus II i don't understand the proof for the limit comparison test

[iScience]

would someone please care to reword this proof for me?

http://en.wikipedia.org/wiki/Limit_comparison_test

it talks about ε, which is not even defined and then n0, which is again not defined, what the hell are all these variables... I'm sure someone here could do a better job organizing that crap proof.

thanks

 

[Stephen Tashi]

You have a minor reason to complain because the proof should say "for each $\varepsilon > 0$ there exists an integer $n_0$..."

The variable $n_0$ isn't "undefined" if you can read mathematical statements such as those found in the definition for the limit of a sequence. In that definition there is a statement about "there exists an integer...". Whatever name is used for that integer isn't an "undefined" variable.

 

[Millennial]

The proof isn't "crap" at all, but it might be a bit difficult to read if you are not used to it. I will try and clarify the proof given by Wikipedia.

First, the proof picks a small positive number $\epsilon$ and says that no matter how small it is, there is always a sufficiently big $N_0$ so that for all $n$ greater than it, $\displaystyle \left|\frac{a_n}{b_n}-c\right|$ will be smaller than $\epsilon$. This is the same thing as saying the ratio of the two sequences converges to c.

Then, it plays around with this expression until it gets some inequality involving the two sequences where it can apply the direct comparison test and conclude that they are either both convergent or both divergent.

 

[iScience]

if the limit of a(n)/b(n) = c, then wouldn't {a(n)/b(n)} - c = 0? i don't understand......

 

[Millennial]

There are some problems with your foundations of limits then. The proof simply makes use of the definition of a limit, also called the epsilon-delta definition. Do you know what the statement $\displaystyle \lim_{n\to\infty} a_n$ rigorously means?

 

[iScience]

is it referring to the series "a(n)" as "n" --> infinity?

 

[iScience]

and i never understood the definition of the limit using the epsilon and delta terms, but i know it basically says that the value of a limit approaches essentially that value with separation of epsilon as epsilon decreases and decreases. is this remotely correct? lol

 

[Millennial]

is it referring to the series "a(n)" as "n" --> infinity?

I'll take that as a no.
You seem to have deeper foundational issues than can be solved on a topic like this, or at least I think so. Can you check your private messages? I sent you a link.

and i never understood the definition of the limit using the epsilon and delta terms, but i know it basically says that the value of a limit approaches essentially that value with separation of epsilon as epsilon decreases and decreases. is this remotely correct? lol

It is very very remotely correct. It is so remotely correct that it is completely useless in proofs like this, but it is still correct. Still, doing calculus with limits without knowing the epsilon-delta definition is, at the very least, ill advised.

 

[Stephen Tashi]

and i never understood the definition of the limit using the epsilon and delta terms, but i know it basically says that the value of a limit approaches essentially that value with separation of epsilon as epsilon decreases and decreases. is this remotely correct? lol

Millenial's diagnosis is correct. You haven't come to grips with the formal definition of limit, so you can't expect to understand proofs that employ that definition.

I'll go further. Your general approach to learning mathematics is wrong. You are only attempting to understand things in intuitive ways. That is OK as a beginning, but you don't understand mathematics until you can deal with concepts as they are actually defined. You are using the liberal arts approach of "Express the definition in your own words". That won't work for you in mathematics because you don't use precise language.

It's common for people to express disdain for "legalistic hair-splitting" but that's exactly the way that formal mathematical proofs are conducted. Advocates for math prefer to emphasize that math is creative, useful, fun etc. However, the truth is that mathematics has a very legalistic aspect. You have to read mathematical definitions lke a lawyer reading a contract to find a loophole. If you avoid the legalistic side, you end up with a completely mangled set of ideas about math.

 

[Mark44]

if the limit of a(n)/b(n) = c, then wouldn't {a(n)/b(n)} - c = 0?

You are confusing the idea of a limit of a ratio with the value of the ratio for a particular n.

For example, let a_n = n, and let b_n = 2n + 1, where n is a positive integer, n ≥ 1.

a₁/b₁ = 1/3
a₂/b₂ = 2/5
a₃/b₃ = 3/7
.
.
.

It's fairly clear (I hope) that $\lim_{n \to \infty}\frac{a_n}{b_n} = 1/2$, but for each specific value of n, a_n/b_n ≠ 1/2, hence a_n/b_n - 1/2 ≠ 0.