Is it true that $ \Im ( \sum_{k=0}^\infty z_k ) = \sum_{k=0}^\infty \Im( z_k)$ and $\Re ( \sum_{k=0}^\infty z_k )= \sum_{k=0}^\infty \Re( z_k) $?

Question

I've thought of the following statements:

Suppose $ z_k(x) = u_n(x) + i \cdot v_n(x) $ is a sequence of functions from $ [a,b] $ to $ \mathbb{C} $. ( $ u_n(x), v_n(x) \in [a,b] \to \mathbb{R} $ ).
Suppose $ \sum_{k=0}^\infty z_k(x) $ converges pointwise.

Is it true that $ \Im ( \sum_{k=0}^\infty z_k(x) ) = \sum_{k=0}^\infty \Im( z_k(x) ) $?
Similarly, is it true that $ \Re ( \sum_{k=0}^\infty z_k(x) ) = \sum_{k=0}^\infty \Re( z_k(x) ) $?
If they are not true in general, then under what criteria are these statements true?

Both of these are true when the sum is finite, but does the result hold for an infinite sum as in the question? ( I assume the answer must have something to do with uniform convergence, but I'm not sure how to carry out the proof )