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Power Expansion (Complex variables)

  1. Jun 22, 2014 #1
    1. The problem statement, all variables and given/known data

    Use the power series for e^z and the def. of sin(z) to check that
    sum ((-1)^k z^(2 k+1))/((2 k+1)!)

    2. Relevant equations



    3. The attempt at a solution

    I apologize, but I am not particularly good with latex. Therefore, I attached a picture of my solution thus far. I've tried many methods, but this is where I get stuck and I can't seem to get sin(z) to equal its power expansion. Any help would be very much appreciated.
     

    Attached Files:

  2. jcsd
  3. Jun 22, 2014 #2
    So you have:
    $$\frac{1}{2i}\sum_{n=0}^{\infty} \frac{z^n}{n!}\left(i^n-(-i)^n\right)$$
    Clearly, if ##n## is even, ##i^n-(-i)^n=0##. Can you figure out what happens if ##n## is odd i.e ##n## is of the form ##2k+1##?
     
  4. Jun 22, 2014 #3
    Yes. Thank you sir!
     
  5. Jun 22, 2014 #4
    Glad to help but please don't call me sir, I am a student myself. :smile:
     
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