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Prove a Sum is Larger than a Root

  1. Feb 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Let nεZ>0. Prove that:

    Ʃ1/√(i) ≥ √(n)

    3. The attempt at a solution
    I'm not sure where to begin, this feels like it should be an induction problem, but I'm not sure how to show that √(n) + 1/√(i+1) ≥ √(n+1). There doesn't seem to be any obvious algebra that would simplify this into what I need it to be.
  2. jcsd
  3. Feb 17, 2012 #2
    Statement correct for n=1.
    For ##n\ge2,##
    [tex]\sum_{i=1}^n\frac1{\sqrt i}\ge\int_1^{n+1}\frac1{\sqrt x}\ dx
    =2\sqrt{n+1}-2\ge\sqrt n.[/tex]
  4. Feb 17, 2012 #3
    No, you have to show that:

    √(n) + 1/√(n+1) ≥ √(n+1).

    In your last step of the induction proces.
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