Maths · Mathematical induction and binomial theorem. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
When n is a positive whole number, (1 + x)n has n + 1 terms and then stops. When n is negative or a fraction, the same pattern of terms goes on for ever. This endless expansion is called the binomial series.
The series only gives the right value when |x| < 1. This is the condition for validity. The coefficients cannot be written with nCr, so you build each one from n(n − 1)(n − 2)... divided by a factorial.
If x is small, the first two or three terms already give a good approximation. You should also know the common series well enough to recognise them when they are written out.
Students put only x into the formula when the bracket is (1 − 2x) or (1 + 3x). Replace x by the whole term, with its sign, and square or cube all of it; the validity condition changes in the same way.
Answer: 1 − 6x + 24x2, valid for |x| < 1/2.
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