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Cauchy sequence

  1. Oct 24, 2007 #1
    How could i show that the sequence
    (An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy?
    Thanks in advance!
  2. jcsd
  3. Oct 24, 2007 #2
    You need to show it is convergent.
    Define [tex]f(x) = \frac{1}{\sqrt{x}}[/tex]. Now confirm that [tex]\sum_{j=1}^{n} \frac{1}{\sqrt{j}} \geq \int_1^n \frac{1}{\sqrt{x}}dx[/tex]. Use this to create a lower bound. Now apply the monotone theorem.
  4. Oct 25, 2007 #3
    Hmm.. I was hoping that there would be a way to do this straight from the definition of a Cauchy sequence, without use of the notion of a definite integral.. thanks though! any other ideas?
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