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Convergence of series using ratio test

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data
    assume summation of series An converges with all An>0. Prove summation of sqrt(An)/n converges

    2. Relevant equations

    3. The attempt at a solution
    I Tried using the ratio test which says if lim as n goes to infinity of |Bn+1/Bn|<1 then summation of Bn converges. I let Bn be Sqrt(An)/n and we have...

    lim as n goes to infinity of |(sqrt(An+1/An)*n/n+1|. limit of n/n+1 goes to one so i need to prove that |sqrt(An+1/An)|<1. But I got stuck because just because An converges does not mean that |(An+1/An)|<1. Can someone help me or suggest on a different convergence test that I should use?
  2. jcsd
  3. Sep 24, 2011 #2


    Staff: Mentor

    For your series,
    [tex]\lim_{n \to \infty} \sqrt{\frac{a_{n+1}}{a_n}} = 1[/tex]

    so the Ratio Test is not going to be any help.

    What other tests do you know?
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