BINOMIAL EXPANSIONS ( 1 + x ) n

( a + b ) n = n C 0 a n + n C 1 a n-1 b + n C 2 a n-2 b 2 + … When n is a positive integer, the binomial expansion gives: From the above, it can be seen that, when a = 1, and b = x: ( 1 + x ) n = n C 0 1 n + n C 1 1 n-1 x + n C 2 1 n-2 x 2 + n C 3 1 n-3 x 3 + … This can be shown to be true for all values of n, and converges if | x | < 1 i.e. – 1 < x < 1. ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 !

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results.Blaise PascalGreek mathematicianEuclidIndian mathematicianPingalaPascal's triangleHalayudhaPersian mathematicianAl-KarajiChinese mathematicianYang Hui

Example 1: Expand as a series in ascending powers of x up to the term in x 2. = ( 1 + 2x ) 1212 = (2x) ( ) 1212 – (2x) 2 + … 2 !2 ! = 1 + x x 2 2 ! – + … = 1 + x – 1212 x2x2 + … ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 !

Example 2: Expand as a series in ascending powers of x up to the term in x 3. 1 ( 1 – 2x ) 3 1 ( 1 – 2x ) 3 = ( 1 – 2x ) –3 = 1 + (–3) (–2x) + (–3)(–4)(–2x) 2 + 2! (–3)(–4)(–5)(–2x) ! = 1 + 6x+ 24x x ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 !

The Binomial Expansion: ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 ! is only valid, when the first term in the bracket is 1. When it is not, we can use the following: ( a + x ) n = a n ( 1 + ) n x a

Example 3: Expand as a series in ascending powers of x up to the term in x 3. 1 ( 2 + x ) 2 1 ( 2 + x ) 2 = ( 2 + x ) –2 = 2 –2 ( 1 + ) –2 x (–2) x 2 ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 ! x 2 ( ) 3 + (–2)(–3)(–4) + (–2)(–3) x 2 ( ) 2 2! 3! = – x 3 16 x2x2 + – 1818 x3x = 1414       This power is often forgotten !!

Example 4: Expand as a series in ascending powers of x up to the term in x 2. Use the expansion to find an estimate of. Give the answer to 3 decimal places. ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 ! 1212 ( ) x 9 – – 1212 x 9 – 2 = ( 9 – x ) 1212 = ( 1 – ) x × 1 2! = 3 If x = 2, – = ≈ 1616 x – – x2x Note, the series is valid if x 9 – 1 < < 1 i.e – 9 < x <   1 + = 3   2626 –

Summary of key points: This PowerPoint produced by R.Collins ; Updated Mar ( 1 + x ) n = 1 + n x + n (n – 1) x 2 + n (n – 1)(n – 2) x 3 + … 2 !2 ! 3 !3 ! This can be shown to be true for all values of n, and converges if | x | < 1, i.e. – 1 < x < 1. The expansion is only valid when the first term in the bracket is 1. When it is not, we can use the following: ( a + x ) n = a n ( 1 + ) n x a The Binomial expansion is given by: Do not forget this power !!