[Solved] MA1971 Exercise Set IV

1. A real number x is called a root number iff x = ⁿ m (that is, x is the n-th root of m) where n and m are positive integers. Please show that there are a countable number of root numbers.
2. Please define the continued fraction

as a sequence, then show that this sequence converges, and find its value. 3. Please define the continued radical

s

x + x + x + x + . . .

as a sequence of functions (expressions), then show that this sequence converges, and find its value as a function of x (expression in x).

4. Please define the continued fraction

,

as a sequence, then show that the odd elements of this sequence oscillate with decreasing amplitude of oscillations, and thus converge to some real number.