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Using Dirichlet's test for uniform convergence

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that [tex]\sum[/tex](-1)^(n+1) / (n + x^2) converges uniformly but not absolutely on R.


    2. Relevant equations
    Using Dirichlet's Test for uniform convergence. (fn) and (gn) are sequences of functions on D satisfying:
    [tex]\sum[/tex] fn has uniformly bounded partial sums
    gn -> 0 uniformly on D
    g(n+1)(x) [tex]\leq[/tex] gn(x) for all x in D and all n in N


    3. The attempt at a solution
    I got parts a) and c) down since obviously (-1)^(n+1) is uniformly bounded by 1 and 1 / (n+1+x^2) < 1 / (n + x^2) for all x in R and all n in N. I'm having trouble showing that 1 / (n+x^2) is uniformly convergent on R. I tried using Weierstrass' M-test but can't seem to make that work.
     
  2. jcsd
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