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Math Help - converge or not???

  1. #1
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    converge or not???

    let \ x_1=a>0 \ and \ x_{n+1} = x_n+ \frac{1}{x_n}, \ n \in N .
     Does \ x_n \ converge \ ?
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  2. #2
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    Quote Originally Posted by flower3 View Post
    let \ x_1=a>0 \ and \ x_{n+1} = x_n+ \frac{1}{x_n}, \ n \in N .
     Does \ x_n \ converge \ ?

    As  a> 0 , it's easy to show inductively that x_n\geq 2\,\,\forall\,2\leq n\in\mathbb{N}.
    If the sequence converged to a finite limit \alpha, which must be \alpha \neq 0 by the above , then using arithmetic of limits we'd get \lim_{n\to\infty}x_n=\lim_{n\to\infty}x_{n-1}+\frac{1}{\lim_{n\to\infty}x_{n-1}}\Longrightarrow\,\alpha=\alpha+\frac{1}{\alpha}  \,\Longrightarrow \alpha=1 , and b the above it's clear that this can't be so the sequence never converges to a finite limit.

    Tonio
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