Question 106
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!
$$
A = 1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \ddots_{+} \frac{1}{2006 + \frac{1}{2007}}}}} \text{ and } B = 1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \ddots + \frac{1}{2005 + \frac{1}{2006}}}}}
$$
Which of the two numbers is greater, $A$ or $B$? Explain your answer!