# converge or not???

• November 22nd 2009, 01:53 AM
flower3
converge or not???
$let \ x_1=a>0 \ and \ x_{n+1} = x_n+ \frac{1}{x_n}, \ n \in N .$
$Does \ x_n \ converge \ ?$
• November 22nd 2009, 02:12 AM
tonio
Quote:

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