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Taylor series of 1/(1 + x^2)

  1. Oct 20, 2011 #1
    The equation starts at B and this is my attempt. As you can see it soon complicates and doesn't look like what t should since I already know what the Taylor series of his function should look like. Is there some clever trick to it that I am missing? PS the series is centred around x = 0.

    [PLAIN]http://img823.imageshack.us/img823/3459/phys.png [Broken]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 20, 2011 #2

    HallsofIvy

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    A Taylor's series about what central point? Rather than work out a large number of derivatives, I would calculate a few derivatives at the given point and try to find a pattern.

    For example, the Taylor's series about x= 0 (the McLaurin series), has f(0)= 1, f'(0)= 0, f''(0)= -2, f'''(0)= 0, f''''(0)= 4!, etc. so I would hypothesize that the nth derivative, at 0, is 0 for odd n, [itex](-1)^n n![/itex] for n odd. Then I would try to prove that is true by induction.
     
  4. Oct 20, 2011 #3
    it's at x = 0, I will try your method and post a pic.
     
  5. Oct 20, 2011 #4

    lurflurf

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    Either write
    1/(1+x^2)=1/(1-(-x^2))
    and expand in geometric series
    or apply Leibniz product rule to
    [(1+x^2)/(1+x^2)]
    and note
    (1+x^2)'''=0
     
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