Math

[Gota]

There is a sentence in math that says that if lim(as n->infinity)of An=L than the lim(as n->infinity) of the arithmetical average of the numbers in An is also L.

The opposite is not true.
If i am asked to show the opposite is not true all i need to do is give an example when it isn't true?or do I need to give some sort of general proof?

If I need a general proof can anyone roughly explain the way to prove it?

[Kloot]

You are given a proof "if A, then B" (A ==> B). You want to show that the opposite "if B, then A" (B ==> A) is not true. You can do that by finding a B such that A is false; in other words, a counter-example. If (and only if) you cannot think of such an example you must prove "if B then not A" in general.

[Gota]

Well i can find an example but how would i go about proving this in general?any ideas?

An example would be (-1)^n when n goes from 1 to infinity.

[Master-Athmos]

You know about induction?

[Gota]

You know about induction?

Yes.
How can it help me in this case?dont think it can.

[koshi]

First of all I think you should be explicit about what you're trying to disprove, ie what the inversion of the theorem (not sentence) really is.

[smoth]

http://www.cut-the-knot.org/proofs/index.shtml

[CarRepairer]

Is capital A a function? I don't understand the notation An if so. I am used to f(n).

[Argh]

Try playing with the value of S.

[Auswaschbar]

Is capital A a function? I don't understand the notation An if so. I am used to f(n).

Guess what he means is A_n (in tex-notation)

[CarRepairer]

Is capital A a function? I don't understand the notation An if so. I am used to f(n).

Guess what he means is A_n (in tex-notation)

Kids these days and their tex. In my day we used mechanical pencils.

[Gota]

An is a sequence.a[n]

[CarRepairer]

I guess you were waiting for an answer... I'm interpreting from your wording that you're given the fact that the if first formula converges to a limit then the second converges to the same limit. Your job is just to prove that if the second formula converges to a limit then the first formula won't necessarily converge to the same limit (it might converge to a different limit or diverge). You gave the example of a[n] = -1^n which converges to a limit in the second but diverges in the first (from a glance anyway). That counterexample is all you need to prove what was asked.

[Tribulex]

spring forum is a great place to get your homework done.

[Gota]

I guess you were waiting for an answer... I'm interpreting from your wording that you're given the fact that the if first formula converges to a limit then the second converges to the same limit. Your job is just to prove that if the second formula converges to a limit then the first formula won't necessarily converge to the same limit (it might converge to a different limit or diverge). You gave the example of a[n] = -1^n which converges to a limit in the second but diverges in the first (from a glance anyway). That counterexample is all you need to prove what was asked.

But what if i wanted to prove it in a general form.how would i go about doing that?

[yuritch]

You can't really prove it in 'general', just because there ARE cases when it's true (so any 'general' proof would be false just because of them). Your example of -1^n is proof enough, you don't need more.

Idea is this: if you want to prove some assumption is false, you only need 1 example of it being false. If you want to prove an assumption is true however, THEN you need a general proof.

(at least those are based on my memories of math courses, that was some years ago so I may be missing something)

[Gota]

Well the definitions of limits use stuff like epsilon so maybe i need to choose a certain epsilon and show there is a contradiction in some way...im just not sure how.This really calls for someone that remembers differential and integral math.

[Dragon45]

state problem exactly - copypasta problem definition with right notation and typset. jpg screenie of the homework sheet the teacher emailed out is fine.

[Gota]

Its all in hebrew...
Ill try to be precise.
I need to show that if the limit of the arithmetical average((a1,a2,a3....an)/n) of the numbers in a general sequence({an} is L(when n runs to infinity) it does not mean that the limit of an(when n runs to infinity) is also L.

[Peet]

Counterexample - an=sin(n). L = limit of (Σai)/n as n goes to infinity = 0.
The limit of an as n goes to infinity does not exist, so it's not equal to L.