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Homework Help: The Fourier Series

  1. Nov 27, 2009 #1
    1. The problem statement, all variables and given/known data

    f(x) = 1 0<t<1
    = -1 1<t<2

    How can I simplify this given that function(on the attachment).

    What I mean is that how can I write the function in any other way?

    In addition, How can I know if the function can be written in other form?
    How can I write the function in other form?

    2. Relevant equations

    3. The attempt at a solution

    Attached Files:

  2. jcsd
  3. Nov 27, 2009 #2


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    Science Advisor

    If n is even, 1- (-1)n= 1- 1= 0! If n is odd, 1- (-1)n= 1- (-1)= 2.

    [tex]\frac{2}{\pi}\sum_{n=1}^\infty \frac{[1- (-1)^n] sin(n\pi t)}{n}[/tex]
    is just
    [tex]\frac{2}{\pi}\sum \frac{2 sin(n\pi t)}{n}[/tex]
    where now the sum runs only over odd n. One way to show that is to use 2n+1 rather than n in the body of the sum. That way, as n goes over all non-negative integers, 2n+1 goes over all positive odd integers:
    [tex]\frac{4}{\pi}\sum_{n=0}^\infty \frac{sin((2n+1)\pi t)}{2n+1}[/tex]
  4. Nov 27, 2009 #3
    Can 2n+1 be 2n-1 provided that n=1 to infinity?
    How can I know if the function can be converted in some form?
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