GCSE-ready- Home
- Lessons
- IGCSE Further Pure Mathematics
- Binomial series for any rational n
Binomial series for any rational n
🎬 The doodle video for this lesson is coming soon. Subscribe on YouTube to see it first.
When n is negative or a fraction, the binomial expansion never stops. It only works when x is small enough.
The infinite series
(1 + x)n = 1 + nx + n(n − 1)2!x2 + n(n − 1)(n − 2)3!x3 + ...
If n is negative or a fraction, the series goes on forever.
It is valid only for |x| < 1.
If n is negative or a fraction, the series goes on forever.
It is valid only for |x| < 1.
Examples
(1 + x)−1 = 1 − x + x2 − x3 + ...
(1 + x)12 = 1 + 12x − 18x2 + ...
The x2 coefficient: (1<span>2)(−12)/2 = −18
(1 + x)12 = 1 + 12x − 18x2 + ...
The x2 coefficient: (1<span>2)(−12)/2 = −18
Validity with kx
(1 + 3x)−1: replace x by 3x. It is valid when |3x| < 1, so |x| < 13.
(1 − 2x)−2 = 1 + 4x + 12x2 + ..., valid for |x| < 12.
(1 − 2x)−2 = 1 + 4x + 12x2 + ..., valid for |x| < 12.
Use the first three terms of (1 + x)12 to estimate √1.02.
- x = 0.02
- 1 + 12(0.02) − 18(0.02)2
- 1 + 0.01 − 0.00005
Answer: 1.00995
Use the same formula for any rational n. The series is infinite unless n is a positive whole number, and (1 + kx)n is valid only for |kx| < 1.
Check you have got it
Answer 7 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.