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.
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?