General Binomial Expansion - Multiple (Edexcel A-Level Mathematics): Flashcards

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

$\binom{n, k} = \frac{n!}{k!(n-k)!}$

$(1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k$

It converges for $|x| < 1$.

The second term is $nx$ for $(1 + x)^n$, $n \geq 1$.

Use multinomial expansion.

$\sum_{k=0}^{n} \sum_{l=0}^{k} \binom{n}{k} \binom{k}{l} x^l y^{k-l}$

To approximate functions using polynomial expressions.

$1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 + ...$