Uniform convergence implies $\sum \int f_n = \int \sum f_n$, what about the the converse?

Question

I have recently read about uniform convergence and I cannot find the answer to this question.

I have read that if the sum is uniformly convergent then your can swap the integrals, but I can't find anywhere if the following is true, or under what conditions it holds:

If a $f_{n}$ is not uniformly convergent on $I \subset \mathbb{R}$ and $a,b \in I$ then $$\int_{a}^{b}\sum_{k=0}^{\infty}f_{k}(x) \neq \sum_{k=0}^{\infty}\int_{a}^{b}f_{k}(x)$$

Is this true in general? And if not, what are the necessary conditions such that the non-equality holds?

Answer 1

Series $$\sum_{n=1}^{\infty}\left[ \frac{nx}{1+n^2x^2}- \frac{(n-1)x}{1+(n-1)^2x^2}\right]$$ have on $[0,1]$ continuous sum, is not uniformly converged and we cannot replace integral with sum.