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Math Help - interchangability of limits and sums

  1. #1
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    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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by oblixps View Post
    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
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  3. #3
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    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?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    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
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