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.