This is not true if [math]p_1 > 0[/math] and [math]q_1 < 0[/math].
[math]p_{m+1} + q_{m+1}\sqrt{r} = \left(p_m + q_m \sqrt{r} \right)\left(p_1 + q_1 \sqrt{r} \right)[/math]
[math]= \left(p_m p_1 + r q_m q_1\right) + \left(p_m q_1 + q_m p_1\right)\sqrt{r}[/math]
If [math]p_m > 0[/math] and [math]q_m < 0[/math], then this means that [math]p_{m+1} > 0[/math] and [math]q_{m+1} < 0[/math]. Thus, by induction, all the [math]p_m[/math] are positive and all the [math]q_m[/math] are negative. Thus, all [math]\frac{p_m}{q_m}[/math] are negative, so the limit o...
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