Investigation

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 about this sequence? $1, \quad 1 + \cfrac{1}{2}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2}}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2}}}, \quad \dotsc$