Term-wise Differentiation of Power Series

[Hootenanny]

For those who don't know I'm writing a tutorial (https://www.physicsforums.com/showthread.php?t=139690") 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.

 

[flebbyman]

I would say that it follows from the linearity of differentiation

 

[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.

 

[Hootenanny]

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?