Bridge to Higher Mathematics, MA 1971 D21
Exercise Set IV
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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.
