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Math Help - def of sequence

  1. #1
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    def of sequence

    Use the definition of convergent to prove :
     {(2n)}^{\frac {1}{n}} \to 1
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  2. #2
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    Quote Originally Posted by flower3 View Post
    Use the definition of convergent to prove :
     {(2n)}^{\frac {1}{n}} \to 1
    Yes, what have you done? What is the definition of convergence for a sequence?

    For this particular problem, you might consider taking the logarithm of every number in the sequence and looking at that new sequence.
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  3. #3
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    proof:
    let \ \epsilon >0 \ want \ to \ prove \  \exists k \in N \  such \ that \ \mid (2n)^{ \frac {1}{n}} \ - 1 \mid < \epsilon ,\  \forall \ n \geq k
    then??
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  4. #4
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     (2n)^{1/n} = 2^{1/n} n^{1/n} . Obviously  2^{1/n} \rightarrow 1 , so we need to only prove that  n^{1/n} \rightarrow 1 .

    For n > 1 we have  n^{1/n} > 1^{1/n} = 1 . So we can let  n^{1/n} = 1 + h_n  where  h_n is some positive quantity depending on n.

    So  n = (1 + h_n)^n . Now, it is clear from the binomial theorem that  (1 + h_n)^n \ge 1 + nh_n + \frac{n(n+1)}{2} h_n \ge \frac{n(n+1)}{2} h_n .

    Can you show that  h_n goes to 0 and so  n^{1/n} = 1 + h_n goes to 1?
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