Question

等比数列｛an｝的首项为1，公比为q，前n项和为S，则数列｛1/an｝的前n项和为？

Answer 1

若q=1,则，a(n) = 1, s(n) = n = s,
1/a(n) = 1, 1/a(1)+1/a(2)+...+1/a(n) = t(n) = n = s.

若q不为1，a(n) = q^(n-1), s(n) = [q^n - 1]/(q-1) = s.
q^n = 1 + (q-1)s.
1/a(n) = (1/q)^(n-1),
t(n) = 1/a(1)+1/a(2)+...+1/a(n) = [1-(1/q)^n]/(1-1/q) = [1-1/(1+qs-s)]/(1-1/q)
= [(qs-s)/(1+qs-s)]/[(q-1)/q]
= sq/(1+qs-s)