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Please help me find examples.

  1. Feb 4, 2008 #1
    1. The problem statement, all variables and given/known data
    Please help me find examples.

    [tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
    [tex]u_{n+1}\leq u_n+v_n[/tex] for all n and [tex]\sum_{n=1}^{\infty}v_n[/tex]
    is finite.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 4, 2008 #2

    morphism

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    Say you take u_n=0 for all n...
     
  4. Feb 4, 2008 #3
    I want to find [tex]v_n\neq 0 ,u_n\neq 0[/tex] for all n
     
  5. Feb 5, 2008 #4

    morphism

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    My advice is to first pick a nice, summable v_n.
     
  6. Feb 5, 2008 #5
    Please help me find examples.

    [tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
    [tex]u_{n+1}\leq u_n+v_n[/tex] for all n and [tex]\sum_{n=1}^{\infty}v_n[/tex]
    is finite.

    In fact, this is a theorem which say that

    [tex]u_n\subset [0,\infty)[/tex] and [tex]v_n\subset [0,\infty)[/tex] such that
    [tex]u_{n+1}\leq u_n+v_n[/tex] for all n
    If [tex]\sum_{n=1}^{\infty}v_n[/tex]
    is finite then [tex]\displaystyle{\lim_{n\rightarrow \infty}u_n}[/tex] exists

    then I want to find [tex]u_n[/tex] which dificult to find lim in order to guarantee
    this therem is well better than MCT because this therem is generalization of MCT.
     
  7. Feb 5, 2008 #6

    morphism

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    Why don't you want to put in some effort?

    Pick a summable (v_n), like say v_n = 1/2^n. Now pick any decreasing (u_n), like u_n = 1/n.
     
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