The rate of change of a bio-reactant R R\,R over a period of time t t\,t minutes is modelled by the function:
R(t)=11t+1(1−3t)(t+2),0≤t<13 R(t) = \frac{11t + 1}{(1 - 3t)(t + 2)}, \quad 0 \le t < \frac{1}{3} R(t)=(1−3t)(t+2)11t+1,0≤t<31Express R(t)R(t)R(t) in partial fractions.
(i) Hence find the binomial series expansion of R(t)R(t)R(t) in ascending powers of ttt, up to and including the term in t2t^2t2. Give each coefficient as a simplified fraction.
(ii) State the range of values of t t\,t for which the expansion in (b)(i) is valid.
