When the expansion never stops: For a positive whole number n the binomial expansion ends. But n can also be negative or a fraction — then it carries on forever as an infinite series (valid when |x| < 1).
IB-style question — first three terms
Find the first three terms of the expansion of (1 + x)⁻¹.
Step by step
- Here n = −1. First term is always 1; second term is nx.
- Third term: n(n − 1)/2! times x².
- Put them together.
Final answer
1 − x + x².
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Replace x with the whole term: For (1 + kx)ⁿ, put kx wherever the formula has x — and remember to square and cube the k as you go.
IB-style question — coefficient inside the bracket
Find the first three terms of the expansion of (1 − 3x)⁻².
Step by step
- Here n = −2 and the 'x' in the formula is replaced by (−3x).
- Second term: n × (−3x).
- Third term: n(n − 1)/2! × (−3x)².
- Put them together.
Final answer
1 + 6x + 27x².