Define

$$
A=1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\ddots_{+} \frac{1}{2006+\frac{1}{2007}}}}} \text { and } B=1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\ddots+\frac{1}{2005+\frac{1}{2006}}}}}
$$

Which of the two numbers is greater, $A$ or $B$ ? Explain your answer!
We will determine the sign of $A-B$. If that number happens to be positive than $A>B$, otherwise $A0$ then $B_{i+1}-A_{i+1}>0$ and vice versa. Thus $A_{1}-B_{1}, A_{3}-B_{3}, \ldots, A_{2005}-B_{2005}$ have the same signs. From

$$
A_{2005}-B_{2005}=\left(2005+\frac{1}{2006+\frac{1}{2007}}\right)-\left(2005+\frac{1}{2006}\right)<0
$$

we get $A<B$.