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Express this series in terms of the given series A(x)

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data

    Let A(x) = a0 + a1x + a2x2 + a3x3 + ... = Ʃanxn.

    Express E(x) = a0 + a2x2 + a4x4 + ... = Ʃa2nx2n.
    Do the same for O(x) = a1x + a3x3 + a5x5 + ...

    2. Relevant equations

    A(x) = a0 + a1x + a2x2 + a3x3 + ... = Ʃanxn

    3. The attempt at a solution
    So I tried using A(x2) as a starting point, but got stuck on how to get the coefficients of A(x2) = a0 + a1x2 + a2x4 + a3x6 + ... to match those of E(x).
    How would I begin expressing E(x) and O(x) in terms of A(x)? Could there be something I could do with sin or cos?
     
  2. jcsd
  3. Jan 13, 2013 #2
    Think about A(-x). What is its series?
     
  4. Jan 13, 2013 #3
    A(-x) = a0 - a1x + a2x2 - a3x3 + ...
    I see now. So when I add A(x)+A(-x), I get 2*E(x). Similarly, A(-x)-A(x) = -2*O(x). Thanks, your hint was a huge help!
     
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