Derivative of an infinite series of functions

Let , . Prove that on (-1,1).

I know that I need to prove uniform convergence on (-1,1). so, Let and for all . I know that converges to 0 and that is always between (-1,1). So using the Dirichlet's test for uniform convergence i should be able to prove is uniformly convergent and therefor . Will this work? and how do i write a formal proof if it is?

Re: Derivative of an infinite series of functions

Uniform convergence of a sequence doesn't guarantee even pointwise convergence of the sequence of derivatives, on the other hand you have the following:

If and for one and uniformly on compact subintervals of then .

In your case take and since the sequence (series) of derivatives is a geometric series the result is immediate.