# Term-wise Differentiation of Power Series

1. Mar 29, 2008

### Hootenanny

Staff Emeritus
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.

2. Mar 29, 2008

### flebbyman

I would say that it follows from the linearity of differentiation

3. Mar 30, 2008

### HallsofIvy

Staff Emeritus
Because you are talking about an infinite series, you also need the fact that a power series converges uniformly inside its radius of convergence.

4. Apr 2, 2008

### Hootenanny

Staff Emeritus
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