Binomial Expansion

The binomial theorem and its generalization for expanding powers of binomials.

Overview

The binomial theorem provides a formula for expanding expressions of the form $(a + b)^n$ without performing repeated multiplication.

Factorial

Definition: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1$$ with $n \in \mathbb{N}$ and $0! = 1$.

Recursive property: $$(n+1)! = (n+1) \times n!$$

Binomial Coefficients

The binomial coefficient is defined as: $$\binom{n}{r} = {}^nC_r = \frac{n!}{(n-r)! , r!}$$ where $n$ is a positive integer, $r \in \mathbb{N}$ and $0 \le r \le n$.

Computational form: $$\binom{n}{r} = \frac{n(n-1)(n-2)\cdots(n-r+1)}{r!}, \qquad n \ge r$$

Properties:

Symmetry: $\binom{n}{r} = \binom{n}{n-r}$
Pascal's identity: $\binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r}$
Special values: $\binom{n}{0} = 1$, $\binom{n}{1} = n$, $\binom{n}{2} = \dfrac{n(n-1)}{2}$

Pascal's Triangle

Triangular array where each number is the sum of the two above it:

        1
      1   1
    1   2   1
  1   3   3   1
1   4   6   4   1
1  5 10 10  5  1

Binomial Theorem (Positive Integer Powers)

For positive integer $n$: $$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$$

Expanded form: $$(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n$$

Characteristics

There are $(n+1)$ terms in total
The first term is $a^n$ and the last term is $b^n$
The power of $a$ decreases by 1 from left to right, while the power of $b$ increases by 1; the sum of the powers in each term equals $n$
The coefficients are symmetrical

General Term

The $(r+1)$th term in $(a + b)^n$: $$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$

This allows finding a specific term without expanding the entire expression.

General Binomial Theorem (Infinite Series)

For any real number $n$ that is not a positive integer (e.g., negative integers, fractions), the expansion of $(1+x)^n$ is an infinite series valid only when $|x| < 1$:

$$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots, \qquad |x| < 1$$

This is called the binomial series. If $|x| \ge 1$, the series diverges.

Finite vs Infinite Expansions

	$n$ positive integer	$n$ not a positive integer
Form	$(1+x)^n = 1 + {}^nC_1 x + {}^nC_2 x^2 + \cdots + x^n$	$(1+x)^n = 1 + nx + \dfrac{n(n-1)}{2}x^2 + \cdots$
Series length	Finite ($n+1$ terms)	Infinite
Validity	Valid for all $x \in \mathbb{R}$	Converges only for $\|x\| < 1$

Approximations Using Binomial Expansion

Case 1: Positive integer powers

For small $b$ relative to $a$, expand $(a+b)^n$ directly and truncate to obtain a decimal approximation.

Example: $(2.02)^6 = (2 + 0.02)^6$

Case 2: Non-integer or negative powers

Rewrite the expression in the form $k(1+x)^n$ where $|x| < 1$, apply the binomial series, and truncate.

Example: $\sqrt[3]{8.064} = \sqrt[3]{8(1+0.008)} = 2(1+0.008)^{1/3}$

Always check that $x$ satisfies the validity condition $|x| < 1$.

Applications

Approximation of expressions and numerical values
Finding specific coefficients or terms
Probability (binomial distribution)
Series expansions in calculus