$$ \lim_{n \rightarrow \infty} \int_{a}^b\ \sum_{k=1}^{n}f_k(x) \mathrm dx= \sum_{k=1}^{\infty}\int_{a}^b f_k(x) \mathrm dx $$
Is this generaly true ? Integral is a sum , two sums can interchange, right?
I ve faced this in a proof of a theorem that says that integral of a uniformly convergent sum is equal to the sum of integral. $\int \sum g_n = \sum \int g_n$ ( $ \sum g_n $ converges uniformly )
The problem i am facing is that the stament in the title is used to prove the previous theorem.
I dont think this is a duplicate, this question is about a partial sum that changes order with an integral not an infinite
It is said the answer below is incorrect, can someone explain why