Math Is Fun Forum

ronn

Guest

Re: Sum of Fourier series

Hi,

That's one way. Did you check the link I gave too?

Yes but I don't see how it can help me.

We end up with:

and

for positive n

I don't understand anything of this. Are you still answering my question or someone else? Please go back and explain how I solve this problem: Find the sum of

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Sum of Fourier series

Yes but I don't see how it can help me.

You don't see that it's relevant to your question? Did you check my post or someone else's?

I don't understand anything of this. Are you still answering my question or someone else? Please go back and explain how I solve this problem: Find the sum of ..

Are you asking because the variable name used is different? Can't you substitute n=2, which is your question and the corresponding answer?

I can't help without you telling specifically what you do not understand. Are you familiar with complex numbers, DeMoivre's formula, integration etc.?

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Sum of Fourier series

Hi;

I don't understand how to use the Fourier series to solve the problem. I don't know what "sgn" and we are not allowed to use terms that we haven't learned to solve problems. Could you, or someone else, please show how you mean that the Fourier series can be used to solve the problem.

I suggest you answer gAr's questions so that he can instruct you on how to do the problem in another way. I will post the solution in terms of Fourier series for both you and ElainaVW.

First, the solution I gave earlier actually shows how to do the problem.

Like in any series work it is important to know a few standard series. We will use two of them.

If we expand x / 2 in a Fourier series we get:

You should know how to do this but I found this on online in case you can not. It is a standard series.

If we expand (π/4) sgn(x) in a Fourier series we get:

this one too can be found on the internet or in any book or at Wolfram alpha. It is a standard series.

If we subtract the second series from the first one we get

we are left with

If we multiply this term by term by -2 we get:

Now you can notice that the RHS is the same as your sum.

All you have to do now is substitute x = 1 into the RHS and that completes the proof.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ronn

Guest

Re: Sum of Fourier series

I can't help without you telling specifically what you do not understand. Are you familiar with complex numbers, DeMoivre's formula, integration etc.?

Yes I know complex numbers, de moivres formula and integration. Can you show how to solve the problem that I wrote now, [comment removed by administrator] instead of referring to another one?

And to bobbym: I know Fourier series also,[comment removed by administrator] the only thing I don't know about what you wrote is "sgn(x)". Also why don't you use the sigma sign? It's easier, clearer and more mathematically correct than to write "+...+"

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Sum of Fourier series

The sigma operator is exactly the same as +...+. It is not more mathematically correct. My way shows how the series is formed. Please wait for the rest of the proof in post #28

sgn(x) is called the sign function. In this case sgn(1)  = 1.

http://en.wikipedia.org/wiki/Sign_function

I know Fourier series also, the only thing I don't know about what you wrote is "sgn(x)"

I don't understand how to use the Fourier series to solve the problem

Based on the above quote, no work, post 8 and 9, I could not assume that you know about the Fourier series.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Sum of Fourier series

Yes I know complex numbers, de moivres formula and integration. Can you show how to solve the problem that I wrote now, [comment removed by administrator] instead of referring to another one?

No, I won't. You are still not specific. If you read the other one, you should have understood the relevance. All that is required is a change of sign.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Sum of Fourier series

The answer I linked to is unnecessarily long. We can solve simply by using Maclaurin series for complex numbers:

Use DeMoivre's theorem and equate real and imaginary parts:

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."