Find the limit of the sequence

In summary, the limit of the sequence a_n = (n^2)(1-cos(5.2/n)) is indeterminate as n approaches infinity. This can be rewritten as a_n = (1-cos(5.2/n))/(1/n^2) and solved using l'Hopital's rule or the Taylor series for cosine.
  • #1
ProBasket
140
0
Find the limit of the sequence whose terms are given by [tex]a_n = (n^2)(1-cos(5.2/n))[/tex]

well as n->inf, cos goes to 1 right? so shouldn't the limit of this sequence be 0?
 
Physics news on Phys.org
  • #2
ProBasket said:
Find the limit of the sequence whose terms are given by [tex]a_n = (n^2)(1-cos(5.2/n))[/tex]

well as n->inf, cos goes to 1 right? so shouldn't the limit of this sequence be 0?

Mathematica returns:

[tex]\mathop \lim\limits_{n\to \infty}n^2[1-Cos(\frac{a}{n})]=\frac{a^2}{2}[/tex]

I'd like to know how too.
 
  • #3
But [tex]n^2[/tex] goes to infinity, so your limit is an indeterminate form, infinity*0. Rewrite as:

[tex]a_n=\frac{1-\cos{\frac{5.2}{n}}}{\frac{1}{n^2}}[/tex]

and use l'hopital, or you could use the taylor series for cos.
 
  • #4
shmoe said:
But [tex]n^2[/tex] goes to infinity, so your limit is an indeterminate form, infinity*0. Rewrite as:

[tex]a_n=\frac{1-\cos{\frac{5.2}{n}}}{\frac{1}{n^2}}[/tex]

and use l'hopital, or you could use the taylor series for cos.

Right, just twice. Thanks.
 

Related to Find the limit of the sequence

What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule.

What is the limit of a sequence?

The limit of a sequence is the number that the terms in the sequence approach as the sequence continues indefinitely.

How do you find the limit of a sequence?

To find the limit of a sequence, you can use various methods such as the squeeze theorem, the ratio test, or the root test. These methods involve analyzing the behavior of the terms in the sequence as they approach infinity.

What is the importance of finding the limit of a sequence?

Finding the limit of a sequence is important in determining the behavior and convergence of a series. It can also help in solving real-life problems that involve continuously changing quantities.

What are some common types of sequences?

Some common types of sequences include arithmetic sequences, geometric sequences, and Fibonacci sequences. These sequences have their own distinct patterns and properties that can be used to find their limits.

Similar threads

  • Introductory Physics Homework Help
Replies
28
Views
478
  • Calculus and Beyond Homework Help
Replies
1
Views
415
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
9
Views
3K
  • Introductory Physics Homework Help
2
Replies
69
Views
3K
  • Calculus and Beyond Homework Help
Replies
8
Views
878
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Science and Math Textbooks
Replies
3
Views
962
  • Calculus and Beyond Homework Help
Replies
1
Views
749
Replies
14
Views
1K
Back
Top