Uniform Convergence: Intuitive Explanation

[stephenkeiths]

I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: \int\sum b_{j}z^{j}dz=\sum\int b_{j}z^{j}dz=\int f(z)dz

On an intuitive level I don't understand why uniform convergence is necessary. I figured that since the integral is linear this is trivial. I was wondering if someone could explain this to me. Maybe elaborate on what can break down, so that they aren't equal if \sum b_{j}z^{j} doesn't uniformly converge to f(z)

 

[HallsofIvy]

Think about this simple example of what non-uniform convergence can do- f_j(x) is 0 if x< 0, x/n if 0\le x\le 1/n, and 1 if x> n. f_j is continuous for all j but the lim of the sequence, f(x)= 0 if x\le 0, 1 if x> 0 is not continuous at x= 0.

As for your remark about the integral being linear: that would tell you that
\int\sum f_j(x) dx= \sum \int f_j(x)dx
for any finite sum, not for infinite sums. Both the integral and infinite sum are defined in terms of limits and it is "uniformity" that allows us to swap limits.

 

[arildno]

To follow up on Halls' comment:
The limit of an infinite sum might, for example, be a non-integrable function, if the convergence is not uniform of the partial sums.