real analysis - Changing order of partial sum and integral all under limit to infinity

$$\lim_{n \rightarrow \infty} \int_{a}^b\ \sum_{k=1}^{n}f_k(x) \mathrm dx= \sum_{k=1}^{\infty}\int_{a}^b f_k(x) \mathrm dx$$

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

Answer

This is true (only after my edits) directly by the linearity of the integral and the definition of infinite sum.

What you wrote is correct - An integral and a (finite) sum can be interchanged with no further conditions. This is exactly the linearity of the integral.