converge or not???

**flower3** · November 22nd 2009, 12:53 AM

$let \ x_1=a>0 \ and \ x_{n+1} = x_n+ \frac{1}{x_n}, \ n \in N .$
$Does \ x_n \ converge \ ?$

**tonio** · November 22nd 2009, 01:12 AM

Originally Posted by flower3

$let \ x_1=a>0 \ and \ x_{n+1} = x_n+ \frac{1}{x_n}, \ n \in N .$
$Does \ x_n \ converge \ ?$

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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