# Fourier Series Question

• Mar 9th 2011, 08:43 PM
Paymemoney
Fourier Series Question
Hi
The following two question i am having trouble finding the fourier series.
Can someone tell me what i have done wrong in question 1?
http://img854.imageshack.us/img854/1623/prob1.jpg
In Question 2 i am stuck after subbing t = 0 and 1 into the intergral. How should this be simplified?

P.S
• Mar 9th 2011, 09:39 PM
CaptainBlack
Quote:

Originally Posted by Paymemoney
Hi
The following two question i am having trouble finding the fourier series.
Can someone tell me what i have done wrong in question 1?
http://img854.imageshack.us/img854/1623/prob1.jpg
In Question 2 i am stuck after subbing t = 0 and 1 into the intergral. How should this be simplified?

P.S

Cos of an odd multiple of $\pi$ is $-1$ and cos of $0$ is $1$

CB
• Mar 10th 2011, 03:49 AM
Paymemoney
Quote:

Originally Posted by CaptainBlack
Cos of an odd multiple of $\pi$ is $-1$ and cos of $0$ is $1$

CB

is that reply to question 1 or 2?
• Mar 10th 2011, 05:05 AM
tonio
Quote:

Originally Posted by Paymemoney
Hi
The following two question i am having trouble finding the fourier series.
Can someone tell me what i have done wrong in question 1?
http://img854.imageshack.us/img854/1623/prob1.jpg
In Question 2 i am stuck after subbing t = 0 and 1 into the intergral. How should this be simplified?

P.S

Why do you do what you do with $a_0$?? Do you hate yourself? Much easier to split the integral into a sum:

$\displaystyle{\int\limits^1_{-1}(1-|t|)dt=\int\limits^0_{-1} (1+t)dt+\int\limits^1_0(1-t)dt=1}$ , and then do the same for $a_n\,,\,b_n$

If this exercise is what I think it is then it's a very nice and beautiful one. You'll get both

the sums of $\displaystyle{\sum\limits^\infty_{n=1}\frac{1}{(2n-1)^2}\,\,and\,\,\sum\limits^\infty_{n=1}\frac{1}{n ^2}}$

Tonio
• Mar 10th 2011, 10:25 AM
CaptainBlack
Quote:

Originally Posted by Paymemoney
is that reply to question 1 or 2?

2

cb
• Mar 10th 2011, 03:26 PM
Paymemoney
quick question how would i simplify the following line:

$a_n = \frac{1}{\pi}[\frac{-cos(1-n)\pi}{2(1-n)} - \frac{cos(1+n)\pi}{2(1+n)} - 0]$
• Mar 10th 2011, 07:25 PM
tonio
Quote:

Originally Posted by Paymemoney
quick question how would i simplify the following line:

$a_n = \frac{1}{\pi}[\frac{-cos(1-n)\pi}{2(1-n)} - \frac{cos(1+n)\pi}{2(1+n)} - 0]$

Since $\cos m\pi =(-1)^m$ we have that

$\displaystyle{a_n=\frac{1}{\pi}\left[\frac{(-1)^n}{2(1-n)}+\frac{(-1)^n}{2(1+n)}\right]=\frac{(-1)^n}{\pi}\cdot\frac{2}{1-n^2}}$

Tonio
• Mar 10th 2011, 08:50 PM
Paymemoney
in this case would i need to determine when n-> even and n-> odd. Because i checked the answers and has it like $\frac{2}{\pi}\sum\limits^\infty_{n=1} \frac{1}{4n^2-1}$
• Mar 11th 2011, 01:44 AM
Paymemoney
if $cos(m\pi) = (-1)^{m}$

then what would

$sin(m\pi)$ = ???
• Mar 11th 2011, 02:38 AM
CaptainBlack
Quote:

Originally Posted by Paymemoney
if $cos(m\pi) = (-1)^{m}$

then what would

$sin(m\pi)$ = ???

Sketch a sine curve and observe its behaviour, if you actually try it is not difficult to answer your own question.

Also if you do not know the value of the sine and cosine at interger multiples of pi you are not ready to study Fourier series and need to repeat trig.

CB
• Mar 11th 2011, 05:06 AM
tonio
Quote:

Originally Posted by Paymemoney
if $cos(m\pi) = (-1)^{m}$

then what would

$sin(m\pi)$ = ???

It is a little weird, imho, to make this kind of trivial questions in very basic trigonometry

if you're dealing with Fourier series...even if you don't remember. and this is already weird,

why don't you check your high school books or, at least, google it?

The sine of any multiple of pi is zero...

Tonio
• Mar 13th 2011, 01:33 AM
Paymemoney
i think my question was not clear, my question was based on what does $sin(1-n)$ equal to.

We know that if $cos(1-n)\pi = (-1)^{1-n}$ so would this apply to $sin(1-n)\pi$ as well.
• Mar 13th 2011, 07:31 AM
CaptainBlack
Quote:

Originally Posted by Paymemoney
i think my question was not clear, my question was based on what does $sin(1-n)$ equal to.

No such term appears in this problem or its solution.

CB
• Mar 13th 2011, 09:12 AM
tonio
Quote:

Originally Posted by Paymemoney
i think my question was not clear, my question was based on what does $sin(1-n)$ equal to.

We know that if $cos(1-n)\pi = (-1)^{1-n}$ so would this apply to $sin(1-n)\pi$ as well.

This continues to be weird: first, as CB already pointed out, $\sin (1-n)$ has nothing to do with do with your problem,

and if you meant $\sin (1-n)\pi$ this again is the sine of an integer multiple of pi

and thus equals zero...

Tonio