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Math Help - Limits of sequences

  1. #1
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    Limits of sequences

    How to prove that 1/2 can't be the limit of the sequence {(3n-2)/(4n+1)} ?

    I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.

    But the problem is could not find an epsilon such that |(3n-2)/(4n+1)-1/2| is greater than or equal to that epsilon.

    Any help would be great..
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  2. #2
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    Re: Limits of sequences

    Quote Originally Posted by Kristen111111111111111111 View Post
    I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.
    If ∃ denotes "there exists" and ∀ denotes "for all", then your version of negation of the limit definition is

    ∃ε ∀n0 ∀n. n > n0 ⇒ |xn - 1/2| ≥ ε

    where xn = (3n-2)/(4n+1). I wrote ∀n because you did not provide a quantifier for n, and by default the quantifier is universal. In fact, the correct negation is

    ∃ε ∀n0 ∃n. n > n0 and |xn - 1/2| ≥ ε.

    In words, there exists an ε such that there exist arbitrarily far elements of the sequence whose distance to 1/2 is at least ε. One way to prove that there exist arbitrarily far elements of the sequence with some property P is to show that all elements possess this property starting from some point. That is, to prove

    ∀n0 ∃n. n > n0 and P(xn) (1)

    we show

    ∃n0 ∀n. n > n0 ⇒ P(xn) (2)

    because (2) is stronger that (1). (Why?)

    To prove (2), note that xn approaches 3/4 from below. Therefore, eventually it is greater than say, the midpoint between 1/2 and 3/4, i.e., 5/8. So, take ε = 5/8 - 1/2 = 1/8 and find an n0 such that xn ≥ 5/8 for all n > n0.
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  3. #3
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    Re: Limits of sequences

    Quote Originally Posted by Kristen111111111111111111 View Post
    How to prove that 1/2 can't be the limit of the sequence {(3n-2)/(4n+1)} ?

    I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.

    But the problem is could not find an epsilon such that |(3n-2)/(4n+1)-1/2| is greater than or equal to that epsilon.

    Any help would be great..
    The limit is 3/4. This is clearly not 1/2.
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  4. #4
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    Re: Limits of sequences

    Well the 2nd one is stronger because the 2nd one can imply the first one..
    Thank you so much for your help
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