Term-wise Differentiation of Power Series

  1. Hootenanny

    Hootenanny 9,681
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    For those who don't know I'm writing a tutorial (Introduction / Summary of Differentiation) in the tutorials forum. I have come to the point of introducing Transcendental functions. I would like to introduce the exponential function first (via the Taylor series) and then present the natural logarithm as it's inverse. Although not entirely necessary, I would like to present a concise proof of term-wise differentiation of power series in the tutorial.

    If anyone knows of a concise online proof, or even better, would be willing to contribute a proof directly, please let me know, either in this thread or via PM.

    Thanks for your time.
     
  2. jcsd
  3. I would say that it follows from the linearity of differentiation
     
  4. HallsofIvy

    HallsofIvy 40,310
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    Because you are talking about an infinite series, you also need the fact that a power series converges uniformly inside its radius of convergence.
     
  5. Hootenanny

    Hootenanny 9,681
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    Is it necessary that both the original and differentiated series converges uniformally, I thought that the original series need only converge?
     
    Last edited: Apr 2, 2008
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