let f_k be a sequence of functions such that each converges to 0 as x approaches 0 for each k.

sum (from 1 to infinite) of limit (as x -> infinite) of f_k = limit (as x -> infinite) of the sum (from 1 to infinite) of f_k.

what are the conditions for when this statement is true? I've browsed through some real analysis textbooks but I can't seem to find a statement of this theorem or proposition.