Limit of sequence question with confusing notation

In summary, the question is asking to find the limit of a sequence defined on an interval, with different expressions for even and odd indices. The attempt at a solution involved evaluating the limits using L'Hopital's rule, but the results were different for the even and odd cases, leading to the conclusion that the sequence does not converge.
  • #1
tomtom690
7
0

Homework Statement


Hello, I don't entirely understand the question here. It is the limit of a sequence, however it is defined on what appears to be an interval, and is different if n is odd or even. Excuse the messy notation here, I don't know how to use LaTeX.

Let the sequence {A_n : n>=1} subset of P(Omega) be defined by:
A_n={( (n^2-1)/3n^3+1 , 4n^2/n^2+1 ] n even
{( 5n^2/(2n^2-3) , (7n^2+5)/3n^2 ] n odd

Study if lim A_n, n-->infinity exists.
In words, the n even case is "the interval open on the left and closed on the right from n squared minus one over three n cubed plus one, to four n squared over n squared plus one."
Its the interval bit which doesn't make any sense to me!

Homework Equations


This is part of an advanced probability course (hence the P(Omega)) if this means anything.


The Attempt at a Solution


Well, first I tried evaluating limits as n goes to infinity on all four expressions, hoping that I would get numbers which would be the same in the odd and even case. I used l'hopitals rule in all cases, and for the evens got an interval of (1/3, 4] and the odds of (5/2, 7/3] which is rediculous, for starters!
Thanks in advance
 
Physics news on Phys.org
  • #2
welcome to pf!

hi tomtom690! welcome to pf! :smile:

(have an infinity: ∞ and an omega: Ω and try using the X2 and X2 icons just above the Reply box :wink:)
tomtom690 said:
Let the sequence {A_n : n>=1} subset of P(Omega) be defined by:
A_n={( (n^2-1)/3n^3+1 , 4n^2/n^2+1 ] n even
{( 5n^2/(2n^2-3) , (7n^2+5)/3n^2 ] n odd

Study if lim A_n, n-->infinity exists.
…
Well, first I tried evaluating limits as n goes to infinity on all four expressions, hoping that I would get numbers which would be the same in the odd and even case. I used l'hopitals rule in all cases, and for the evens got an interval of (1/3, 4] and the odds of (5/2, 7/3] which is rediculous, for starters!

well, half the sequence converges to one point in P(Ω), and the other half converges to a different point in P(Ω), so … ? :smile:
 

FAQ: Limit of sequence question with confusing notation

What does "lim n→∞ an = L" mean in limit of sequence notation?

The notation "lim n→∞ an = L" means that as the index n approaches infinity, the sequence an converges to the limit L. In other words, the terms of the sequence get closer and closer to the value L as n gets larger.

How do I determine the limit of a sequence with confusing notation?

The first step is to rewrite the notation in a more familiar form, such as "lim x→a f(x) = L". Then, you can use various methods such as direct substitution, factoring, or L'Hopital's rule to evaluate the limit.

What is the difference between the limit of a sequence and the limit of a function?

The limit of a sequence is the value that the terms of the sequence approach as the index increases without bound, while the limit of a function is the value that the function approaches as the input approaches a specific value. In other words, the limit of a sequence focuses on the behavior of the terms, while the limit of a function focuses on the behavior of the entire function.

Can a sequence have more than one limit?

No, a sequence can only have one limit. This is because the limit of a sequence is a unique value that the terms of the sequence converge to, and if there were multiple limits, the terms would not converge to a single value.

What is the significance of the limit of a sequence?

The limit of a sequence is an important concept in mathematics and is used to describe the behavior of a sequence as the index increases without bound. It can also be used to determine the convergence or divergence of a sequence, and to evaluate more complex mathematical expressions involving sequences.

Similar threads

How to show that these sequences are summable?
Replies
1
Views
1K
A question about definite integrals and series limits
Replies
12
Views
2K
Understanding the Use of Min in Cauchy Sequences
Replies
1
Views
970
Proving convergence of rational sequence
Replies
2
Views
895
Does each norm on vector space become discontinuous when restricted to S^1?
Replies
1
Views
947
Convergence of a recursive sequence
Replies
3
Views
1K
Special Comparison Test For Infinite Series
Replies
4
Views
713
Converging Vs diverging sequences
Replies
4
Views
1K
Show that a recursive sequence converges
Replies
4
Views
1K
##\lim_{n \to \infty} ## for the sequence at ##a_n##
2
Replies
51
Views
4K

Hot Threads

Recent Insights

Back
Top