Infinite Series: ∑ (-1)^(n+1) cos(nx)

[kingwinner]

Homework Statement

Consider the infinite series
∞
∑ (-1)ⁿ⁺¹ cos(nx)
n=1
Is there any convergence?
(Answer: No convergence)

2. Homework Equations /concepts
Infinite series

The Attempt at a Solution

I was looking back in my 1st year calculus textbook, but none of the theorems seem to apply.

Basic divergence test:
If a_n does not converges to 0 as n->∞, then ∑ a_n diverges.
[well...here the a_n's are constants and does not depend on x, so this theorem does not apply in our case]

Secondly, it is not a power series (it has cos(nx) here), so none of the theorems about power series apply here.

Can someone please explain why the above infinite series has no convergence?
Thank you!

 

[Office_Shredder]

Basic divergence test:
If a_n does not converges to 0 as n->∞, then ∑ a_n diverges.
[well...here the a_n's are constants and does not depend on x, so this theorem does not apply in our case]

Why doesn't this apply? For what possible values of x can the series converge, using this theorem?

 

[kingwinner]

Why doesn't this apply? For what possible values of x can the series converge, using this theorem?

Because it says that the form of the series has to be ∑ a_n, and I don't see any x in it. (maybe I am wrong?)

When there is a x in the serires, it should be something like ∑ a_nxⁿ, so the a_n are just constant coefficients (depends on n only, not on x)...

How can I tell for what possible values of x can the series converge?

 

[tnutty]

You know that cos(nx) will always be less than or equal to 1 , so ...

 

[kingwinner]

You know that cos(nx) will always be less than or equal to 1 , so ...

But why does this imply that there is no convergence?

 

[Office_Shredder]

You can say: Pretend x is a constant. Let a_n= (-1)ⁿ⁺¹cos(nx). Does ∑a_n converge? The answer in general will depend on x.

 

[kingwinner]

You can say: Pretend x is a constant. Let a_n= (-1)ⁿ⁺¹cos(nx). Does ∑a_n converge? The answer in general will depend on x.

OK, but how can I possibly test convergence for ALL values of x?

 

[Petek]

OK, but how can I possibly test convergence for ALL values of x?

Can you find a specific value of x for which the series obviously converges? What about a specific value for which it obviously diverges? If so, then are there other such values of x? Finally, can you generalize?

HTH

Petek

 

[kingwinner]

Can you find a specific value of x for which the series obviously converges? What about a specific value for which it obviously diverges? If so, then are there other such values of x? Finally, can you generalize?

HTH

Petek

I can see that for example when x=0, the sequence alternates between -1 and 1, so it does not tend to 0 and so the series diverges.
But there are infinitely many possibilites for x (x can be ANY real number), and I don't see how every possible case can be ruled out...

 

[Office_Shredder]

Think of it the other way: What condition is necessary for the series to converge? Can any x satisfy this condition?

 

[kingwinner]

Think of it the other way: What condition is necessary for the series to converge? Can any x satisfy this condition?

I am sorry, but I don't get it...can you please give me the names of the theorems that can answer your question?

Here is my try:
In order to converge, the sequence of partial sums must converge to a finite number.
But I don't see how this is going to help.

Once again, the trouble that I am having is that I don't know how to deal with the infinitely many possibilites for x...how can I test convergence in each case? How many cases in total are there?

 

[Petek]

I am sorry, but I don't get it...can you please give me the names of the theorems that can answer your question?

He's referring to the test that you gave in your very first post:

Basic divergence test:
If a_n does not converges to 0 as n->∞, then ∑ a_n diverges.

Next, the answer given above ("no convergence") is incorrect, because the series does converge for some values of x. What happens if cos (nx) = 0 (for all n)? For which values of x is this true?

Here is my try:
In order to converge, the sequence of partial sums must converge to a finite number.
But I don't see how this is going to help.

Once again, the trouble that I am having is that I don't know how to deal with the infinitely many possibilites for x...how can I test convergence in each case? How many cases in total are there?

Two. The one given above (cos (nx) = 0 (for all n)) and ?. For the second case (you supply the condition), use what you called the "Basic divergence test".

Please post again if you would like more help.

Petek

 

[kingwinner]

Next, the answer given above ("no convergence") is incorrect, because the series does converge for some values of x. What happens if cos (nx) = 0 (for all n)? For which values of x is this true?

Please post again if you would like more help.

Petek

But I don't think it is possible to have a value of x for which cos(nx)=0 for ALL n. For example, x=pi/2 doesn't work. x=3pi/2 doesn't work as well...

 

[Dick]

But I don't think it is possible to have a value of x for which cos(nx)=0 for ALL n. For example, x=pi/2 doesn't work. x=3pi/2 doesn't work as well...

No, there isn't such a value. But you are on the right track. Now can you take that one step further and show there is no value of x such that limit n->infinity cos(nx)=0? That's really what you need to show nonconvergence, right? It's pretty easy if x is a rational multiple of pi. If it's not, you need a 'real analysis' type theorem, I think. Like if x is irrational then nx mod 1 is a dense set in the interval [0,1]. Is this a real analysis course? Have you proved anything like that?

 

[kingwinner]

No, there isn't such a value. But you are on the right track. Now can you take that one step further and show there is no value of x such that limit n->infinity cos(nx)=0? That's really what you need to show nonconvergence, right? It's pretty easy if x is a rational multiple of pi. If it's not, you need a 'real analysis' type theorem, I think. Like if x is irrational then nx mod 1 is a dense set in the interval [0,1]. Is this a real analysis course? Have you proved anything like that?

No, it's from an applied partial differential equations course. Please explain in the simplest way since I don't have any background in real analysis so far...

In the text, it is trying to justify why the sin Fourier series for x
i.e. x= ∑ (-1)ⁿ⁺¹ (2/n) sin(nx)
cannot be differentiated term-by-term.

So the idea is if
∞
∑ (-1)n+1 cos(nx)
n=1
diverges for all values of x, then the sin Fourier series cannot be differentiated term-by-term, am I right??

 

[Dick]

If x=p*pi/q where p and q are integers (i.e. a rational times pi) then cos(nx)=cos(a multiple of pi) whenever n is divisible by q. That's +1 or -1. In the case where x=r*pi where r is irrational then you can show nx is 'very close' to being a multiple of pi an infinite number of time, so cos(nx) can't approach zero. Maybe they really expect you to prove that, and you can just get a away with stating it and waving your hands around.

 

[Petek]

But I don't think it is possible to have a value of x for which cos(nx)=0 for ALL n. For example, x=pi/2 doesn't work. x=3pi/2 doesn't work as well...

Quite correct. Sorry for misleading you. To make up for my error, here's a simple proof that cos (nx) doesn't converge to 0 for any value of x:

First, recall the following trig identity:

cos (u) - cos (v) = -2 sin \frac{1}{2} (u + v) sin \frac{1}{2} (u - v)

Now suppose that cos (nx) converges to 0 for some value of x. Then cos ((n + 2)x) also converges to 0. Hence cos ((n + 2)x) - cos (nx) converges to 0. By the above trig identity,

cos ((n + 2)x) - cos (nx) = -2 sin ((n + 1)x) sin x

Therefore, either sin ((n + 1)x) converges to 0, or sin x = 0. However, since

cos^2 ((n + 1)x) + sin^2 ((n + 1)x) = 1

we can't have both cos ((n + 1)x) and sin ((n + 1)x) converging to 0. Therefore, sin x = 0. This implies that x = m\pi for some integer m. But then cos (nx) = cos nm\pi = \pm 1, contradicting the assumption that cos (nx) converges to 0.

HTH

Petek