Infinite Limit: How to Prove a Sequence Goes to Infinity

[The_Iceflash]

Homework Statement

Ok. For this sequence: $$a_{n} = n^2\left|cosn\pi\right|$$, Show/Prove that $$a_{n} \rightarrow\infty$$

Homework Equations

N/A

The Attempt at a Solution

I have to manipulate the statement to show that
$$n^2\left|cosn\pi\right| > ?$$

I'm having trouble making a statement that's smaller. If it was a fraction I could do it:

Ex: Manipulating $$\frac{n^3}{n^2+2}$$ gets

$$\frac{n^3}{n^2+2} > \frac{n^3}{n^2+n^2} = \frac{n^3}{2n^2} = \frac{n^3}{2}$$

I replace 2 with n^2 to make the denominator bigger thus making it smaller. I somehow have to do something like that with the sequence given but I'm not sure how.

Any help is appreciated.

 

[Mark44]

I think something is missing in the problem statement. What is the exact wording of this problem?

 

[The_Iceflash]

I think something is missing in the problem statement. What is the exact wording of this problem?

I just added it above.

 

[Mark44]

$$a_{n} \rightarrow\infty$$ > ?

Prove that it's greater than what? That still doesn't make any sense.

 

[The_Iceflash]

Prove that it's greater than what? That still doesn't make any sense.

That's what I have to figure out. The example with the fraction I showed is what I have to do somehow.

 

[Mark44]

Assuming n is an integer, |cos(n*pi)| = 1, so n²|cos(n*pi)| = n². What does that do if n gets large?

 

[The_Iceflash]

Assuming n is an integer, |cos(n*pi)| = 1, so n²|cos(n*pi)| = n². What does that do if n gets large?

If n gets large then the sequence gets large.

So, would n²|cos(n*pi)| > n|cos(n*pi)| be sufficient?

After I would do that I need to set it greater than M and solve for it.

For example: the fraction above I set $$\frac{n^{3}}{2} > M$$ and solved for n.

Then I'd put it all together formally.

 

[Mark44]

Clearly, from the earlier work, $$\lim_{n \to \infty} a_n = \infty$$

Now, to prove that this is so, you want for find a number M s.t., for all n >= M, a_n > M.

You have a_n = n² (reason given earlier) and you want a_n > M.

Can you combine these and solve for n?