• Support PF! Buy your school textbooks, materials and every day products Here!

Limit of a sequence 2

  • Thread starter arl146
  • Start date
  • #1
343
1

Homework Statement


the sequence is an = (cos(n))^2 / 2^n


Homework Equations


none really


The Attempt at a Solution


like i mentioned in my last post, i usually use l'hopitals or dividing by the largest exponent from the denominator. here, i dont see why i would want to use l'hopitals, so thats out of the question. i cant really divide by the largest exponent. is there a way i could break this down or something?
 

Answers and Replies

  • #2
Curious3141
Homework Helper
2,843
86

Homework Statement


the sequence is an = (cos(n))^2 / 2^n


Homework Equations


none really


The Attempt at a Solution


like i mentioned in my last post, i usually use l'hopitals or dividing by the largest exponent from the denominator. here, i dont see why i would want to use l'hopitals, so thats out of the question. i cant really divide by the largest exponent. is there a way i could break this down or something?
What's the maximum value the numerator can take?
 
  • #3
343
1
1. so that means the bottom is always going to be larger.
can i just write on my hw like ..

(cos(n))^2 < 1, for all n
as n approaches infinity, 2^n gets large.
small # / large # = 0
 
  • #4
Curious3141
Homework Helper
2,843
86
1. so that means the bottom is always going to be larger.
can i just write on my hw like ..

(cos(n))^2 < 1, for all n
as n approaches infinity, 2^n gets large.
small # / large # = 0
Yes, although I would write [itex]\cos^2 n \leq 1 \forall n \in \mathbb{N}[/itex]. The strict inequality is actually true because the only values of n that give you a value of exactly one are zero and multiples of [itex]\pi[/itex] and you're dealing with natural numbers here, but proving it is not worth the trouble, and is unnecessary here.

So just write that, and [itex]\lim_{n \rightarrow \infty} 2^n = \infty[/itex], so the quotient tends to zero.
 

Related Threads for: Limit of a sequence 2

  • Last Post
Replies
6
Views
795
  • Last Post
Replies
1
Views
954
  • Last Post
Replies
9
Views
5K
Replies
4
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
9
Views
568
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
11
Views
2K
Top