Bridge to Higher Mathematics, MA 1971
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
1
3 + 1
3− 1
3− 1
3−. . .
as a sequence, then show that this sequence converges, and find its value.
3. Please define the continued radical
s
x +
r
x +
q
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
1
2 + 1
2− 1
2+ 1
2−. . .
,
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.
