Creating a converging series of functions

Question

I have to create a series of functions ($\sum_{k=0}^{\infty}f_{k}$) such that it converges uniformly in $[1,2], [3,4]$ and diverges in $(2,3)$. I tried creating two different power series and taking their sum but I can't promise convergence that way (if one converges in [1,2] and the second in [3,4] it doesn't promise me convergence). So I'm stuck and I don't have an idea how to approach this question. Every hint would help.

Answer 1

What about defining for all $k \in \mathbb{N}$ $$f_k(x)=1 \text{ if } x \in (2,3) \quad \quad \text{and} \quad \quad f_k(x)=0 \text{ otherwise} \quad ?$$