(A Level) Binomial expansion for rational n

Learning objectives

Expand (1 + x)ⁿ for a negative or fractional index n in ascending powers of x, usually as far as the term in x² or x³, simplifying each coefficient

State the set of values of x for which an expansion is valid, starting from the condition |x| < 1

Rewrite (a + bx)ⁿ as aⁿ(1 + bx/a)ⁿ before expanding, and deduce that the expansion is valid for |bx/a| < 1

Expand products and quotients such as (2 − x)/(1 + 3x)², or a sum of partial fractions, by multiplying or adding series and collecting terms up to a given power