# Term-wise Differentiation of Power Series

by Hootenanny
Tags: differentiation, power, series, termwise
 Emeritus Sci Advisor PF Gold P: 9,789 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.
 P: 28 I would say that it follows from the linearity of differentiation
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,713 Because you are talking about an infinite series, you also need the fact that a power series converges uniformly inside its radius of convergence.
Emeritus
Sci Advisor
PF Gold
P: 9,789

## Term-wise Differentiation of Power Series

 Quote by HallsofIvy 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?

 Related Discussions Calculus & Beyond Homework 7 Calculus 3 Calculus & Beyond Homework 4 Calculus 2 Introductory Physics Homework 3