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

What is the binomial expansion for positive integer $n$?

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

How is the binomial coefficient calculated?

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

What is the general binomial expansion for non-integer $n$?

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

When does the binomial series converge?

It converges for $|x| < 1$.

What is the first term in $(1 + x)^{1/2}$ expansion?

What is the second term in the binomial expansion of $(1 + x)^n$?

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

How to handle multiple terms in binomial expansion?

Use multinomial expansion.

What is the form for expanding $(1 + x + y)^n$?

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

What is the purpose of the Taylor series in calculus?

To approximate functions using polynomial expressions.

What is the binomial expansion for $(1 + x)^{1/2}$?

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