Solve Limit of Sequence: Determine Convergence/Divergence

In summary, the sequence (cos[n])^2 / (2^n) can be evaluated using either L'Hospital's Rule or the Squeeze Theorem. However, the Squeeze Theorem is the correct method as the derivative of 2^n is not 2 and the limit of sin(n) as n approaches infinity does not exist. Therefore, the function approaches 0 as n approaches infinity.
  • #1
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Homework Statement


Determine whether the sequence converges or diverges. If it converges, what does it converge to?

(cos[n])^2 / (2^n)

Homework Equations


L'Hospital's Rule
or
Squeeze Theorem


The Attempt at a Solution


As n-->infinity, this function approaches infinity/infinity. Applying L'Hospital's rule gives -(sin[n])^2 / 2 which gives infinity/2=infinity.

However, if I use the squeeze theorem to say that 0<(cos[n])^2<1 and then divide everything by 2^n, then I can say that this function approaches 0.

Which is the correct method?
 
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  • #2
The derivative of [tex]2^{n} = ln(2)*2^{n}[/tex], and the derivative of [tex]cos^{2}(n) = -2cos(n)sin(n).[/tex]
 
Last edited:
  • #3
i) The sequence isn't infinity/infinity. You just said 0<=cos(n)^2<=1. How can it be infinity/infinity?? So you can't use l'Hopital. 2) The derivative of 2^n is NOT 2. The squeeze argument is correct.
 
  • #4
I would also point out that [itex]\lim_{n\rightarrow \infty} sin(n)[/itex] is NOT infinity. It just doesn't exist.
 

Related to Solve Limit of Sequence: Determine Convergence/Divergence

1. What is a sequence in mathematics?

A sequence in mathematics is a list of numbers arranged in a specific order. It can be finite or infinite and each number in the sequence is called a term.

2. How do you determine the convergence or divergence of a sequence?

To determine the convergence or divergence of a sequence, you need to find the limit of the sequence. This is done by taking the limit as n approaches infinity of the nth term in the sequence. If the limit exists and is a finite number, the sequence is convergent. If the limit does not exist or is infinite, the sequence is divergent.

3. What is the difference between a convergent and a divergent sequence?

A convergent sequence is one that approaches a finite limit as the terms in the sequence increase. A divergent sequence is one that does not have a limit or approaches infinity as the terms in the sequence increase.

4. What are some common methods for solving the limit of a sequence?

Some common methods for solving the limit of a sequence include using algebraic manipulation, applying known limit formulas, using the squeeze theorem, and using the ratio test. It is important to carefully analyze the sequence and choose the most appropriate method for solving the limit.

5. Why is it important to determine the convergence or divergence of a sequence?

Determining the convergence or divergence of a sequence is important because it helps us understand the behavior of the terms in the sequence as they approach infinity. This information is useful in many areas of mathematics, such as calculus, where the concept of limits is essential for solving problems.

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