Prove some laws of sequencing limits

In summary: I think you might be misunderstanding something here. The proof of #2 does not rely on the concept of a cauchy sequence. It relies on the fact that for any &epsilon; > 0 there exists an N such that n > N implies |xn-L|<&epsilon;. So if you can show that this holds for all L, then the proof is complete. If you can't do this, then the proof is not complete and you might need to look at other methods. 2. Thanks for the input, but I think I can handle it.
  • #1
fangtu
9
0
Hi, I got some problems with the homework that I was assigned last week. I really got some problems to even start doing it...it's too difficult that I don't even know how to start it...the professor asked us to proof some laws...of sequencing limits when I don't even understand anything...I've discussed the problem with lots of classmates but non of them know how to do them either...please help me PROOF these problems><" thanxz in advance!
PROOF:
#1 Limit Law: lim (anbn) = (lim an ) (lim bn)
n->infinity n->infinity n->infinity


#2 an = (-1)^n (1,-1,1,-1,1,-1...) when n=0,1,2,3... Proof that it diverges
THANKS A LOT IF YOU COULD HELP ME!@@"
 
Physics news on Phys.org
  • #2
I think this is an exercise in 'epsilontics'.
Meaning, you use the standard definition of a limes:

a = lim(n->[oo])an

if and oly if

for each epsilon > 0
there exists n0, so that
|an - a| < epsilon
for each n > n0.
 
  • #3
yeah, I saw my professor writing something like thaht on the board, but I have no idea how and what that is@@"...could you help me doing the problem further?? please?
 
  • #4
I've really never learned that...please help me
 
  • #5
You just saw it? And didn't take notes?

OK, as for #1:
You want to show that |anbn - ab|
goes to zero, OK?
Here's a trick: Let's insert - abn + abn (which is zero):
|anbn - ab|
= |anbn - abn + abn- ab|
= |(an - a)bn + (bn - b)a|

since (an - a) and (bn - b) go to zero, the whole expression goes to zero. It's a matter of writing it down properly.
 
  • #6
oh...ok, thanxz a lot arcnets...actually I did take notes...but the problem is that the professor just went through the whole thing whithout saying much about it...anyways...thank you!
uhmmm...do you have any clue on how to proof number two by chance? Anybody!?@@"
 
Last edited:
  • #7
the first problem looks something like this:
img11.gif

does anyone know how to proof it@@? please?! I desperately want to know it...thanxz
 
  • #8
PLEASEEEEEEEEEEEEE!
 
  • #9
1. arcnets already showed you how to do this, why do you keep asking about it?

2. A sequence converges to a number L if for every &epsilon; > 0 there exists an N such that n > N implies |xn-L|<&epsilon;. Given xk=(-1)k, for any L there exists infinitely many m such that |xm-L| >= 1, so the definition of convergence can't be satisfied satisfied. This prove is a little more slick if you can use the concept of a cauchy sequence which is not a difficult concept.
 
  • #10
thank you grady!
 

1. What are sequencing limits?

Sequencing limits refer to the maximum or minimum values that a sequence can approach as its terms become infinitely large or small.

2. What is the definition of a limit in sequencing?

A limit in sequencing is the precise value that a sequence approaches as its terms become infinitely large or small. It represents the behavior of the sequence as a whole.

3. How do you prove the laws of sequencing limits?

The laws of sequencing limits can be proved using mathematical techniques such as the epsilon-delta method or the sandwich theorem. These methods involve manipulating the terms of the sequence to show that it approaches the desired limit.

4. What are the basic laws of sequencing limits?

The basic laws of sequencing limits include the sum, difference, product, and quotient rules. These rules allow for the manipulation of sequences to find their limits and are essential in proving more complex laws of sequencing limits.

5. Why are the laws of sequencing limits important in mathematics?

The laws of sequencing limits are important because they provide a foundation for understanding and solving problems in calculus and other areas of mathematics. They allow for the evaluation of complicated functions and the analysis of the behavior of sequences and series.

Similar threads

Replies
7
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
Replies
6
Views
525
  • Introductory Physics Homework Help
Replies
2
Views
887
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Introductory Physics Homework Help
Replies
9
Views
3K
  • Calculus and Beyond Homework Help
Replies
13
Views
899
  • Calculus and Beyond Homework Help
Replies
23
Views
1K
Replies
2
Views
335
Replies
13
Views
1K
Back
Top