# interchangability of limits and sums

• February 11th 2011, 11:37 AM
oblixps
interchangability of limits and sums
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.
• February 12th 2011, 12:37 AM
FernandoRevilla
Quote:

Originally Posted by oblixps
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 $\lim_{x \to 0}f_k(x)=0$ with the behaviour of $f_k$ as $x\to +\infty$ ?

If you meant $\lim_{x \to +\infty}f_k(x)=0$ then, the left side of the equality is $0$ and the the answer could be very general depending on the hypothesis we add to the $f_k$ functions.

Fernando Revilla
• February 12th 2011, 12:33 PM
oblixps
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?
• February 12th 2011, 11:01 PM
FernandoRevilla
I would insist that it is a very general problem. For example, is the domain of $f_k$ compact?, are $f_k$ continuous? , etc.

Fernando Revilla