Bài giải :

$\frac{{{10}^{2005}}+1}{{{10}^{2006}}+1}=1-\frac{{{10}^{2006}}-{{10}^{2005}}}{{{10}^{2006}}+1}$

$\frac{{{10}^{2004}}+1}{{{10}^{2005}}+1}=\frac{10\left( {{10}^{2004}}+1 \right)}{10\left( {{10}^{2005}}+1 \right)}=\frac{{{10}^{2005}}+10}{{{10}^{2006}}+10}=1-\frac{{{10}^{2006}}-{{10}^{2005}}}{{{10}^{2006}}+10}$

Vì   $\frac{{{10}^{2006}}-{{10}^{2005}}}{{{10}^{2006}}+1}>\frac{{{10}^{2006}}-{{10}^{2005}}}{{{10}^{2006}}+10}$  nên  $\frac{{{10}^{2005}}+1}{{{10}^{2006}}+1}<\frac{{{10}^{2004}}+1}{{{10}^{2005}}+1}$.