What has to do the hypothesis with the behaviour of as ?
If you meant then, the left side of the equality is and the the answer could be very general depending on the hypothesis we add to the functions.
Fernando Revilla
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.
What has to do the hypothesis with the behaviour of as ?
If you meant then, the left side of the equality is and the the answer could be very general depending on the hypothesis we add to the functions.
Fernando Revilla
Correction:
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 -> 0) of f_k = limit (as x -> 0) of the sum (from 1 to infinite) of f_k.
so in general what kind of sequence of functions would satisfy this property? is it only required that the sequence of functions converges uniformly?
I would insist that it is a very general problem. For example, is the domain of compact?, are continuous? , etc.
Fernando Revilla