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Term-wise Differentiation of Power Series

by Hootenanny
Tags: differentiation, power, series, termwise
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Hootenanny
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Mar29-08, 01:47 PM
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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.
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flebbyman
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Mar29-08, 05:10 PM
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I would say that it follows from the linearity of differentiation
HallsofIvy
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Mar30-08, 06:53 AM
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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.

Hootenanny
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Apr2-08, 08:21 AM
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Term-wise Differentiation of Power Series

Quote Quote by HallsofIvy View Post
Because you are talking about an infinite series, you also need the fact that a power series converges uniformly inside its radius of convergence.
Is it necessary that both the original and differentiated series converges uniformally, I thought that the original series need only converge?


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