For a real number r, let s(r) denote the set of real numbers that can be expressed in the form

a1 r + a2 r^2 + a3 r^3 + ...

with an infinite number of terms, where ai in {0,1}

(a) Show that 1/2 an element in S(1/4)

(b) Show that 1/5 is not an element in S(1/3)

(c) Show that 1/7 is an element in S(1/2)

(d) Show that 1/11 is an element in S(2/3)

(e) Show that 1/13 is not an element in S(1/2)

(f) Show that 1/2 is an element in S(1/4)

(g) Show that 1/4 is not an element in S(1/5)

(h) Show that 1/5 is an element in S(1/4)

(i) Show that 1/4 is not an element in S(1/7)

Guest Sep 1, 2022