Consider the sequence \[1, \quad 1 + \cfrac{1}{1}, \quad 1 + \cfrac{1}{1 + \cfrac{1}{1}}, \quad 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}, \quad \dotsc \]

What do you make of it?

What about this sequence? \[1, \quad 1 + \cfrac{1}{2}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2}}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2}}}, \quad \dotsc \]

What other sequences might you try?

What questions would you ask about these sequences?