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Application of the Cauchy product

  1. Jun 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Hello,

    I'm trying to find the Taylor representation of a product of functions - the exponential of x times sin(y). Also for (x - y)sin(x+y).

    3. The attempt at a solution

    Well, I want to use the Cauchy product in both cases.

    I know the taylor representation of both functions:

    Ʃ x^n / n! = e^x (a_n)

    Ʃ (-1)^k * y^(2k+1) / (2k+1)! = sin(y) (b_k)

    (infinite sums)

    so:

    Ʃ x^n / n! * Ʃ (-1)^k * y^(2k+1) / (2k+1)!

    the Cauchy product:

    ƩƩ [ x^n / n! * (-1)^(n-k) * y^(2(n-k)+1) / (2(n-k)+1) ] (Both sums got to infinity)

    Here I'm stuck. I don't know what I could do to simplify that expression. Any suggetions?

    For the other example, is it valid to use the taylor representation of the sine function and just substitute the argument with (x+y) ? And then multiply the (x-y) with the sum?

    Thanks!
     
    Last edited: Jun 21, 2012
  2. jcsd
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