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Homework Help: Limit of product of sequences

  1. Jun 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]



    2. Relevant equations

    sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1

    3. The attempt at a solution

    I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).

    I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. I need to do an epsilon proof of the limit and I am not sure how to solve the equation in 2. in terms of epsilon. Thanks for your help.
     
  2. jcsd
  3. Jun 8, 2010 #2
    You can't find a limit using epsilon/delta.

    Evaluating the limit is almost immediate after simple algebra [tex](a+b)(a-b)=a^2-b^2[/tex], and yes the limit is 1/2.

    If you still want to prove it by definition you should for any given [tex]\epsilon>0\ find\ N>0\ so\ \forall\ n>N\ |a_n-1/2|<\epsilon [/tex]
     
    Last edited: Jun 8, 2010
  4. Jun 8, 2010 #3

    vela

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    What do you mean "how to solve the equation in 2 in terms of epsilon"? You can get that equation using estro's hint and use it to show that the limit of the sequence is indeed equal to 1/2 as n goes to infinity.
     
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