Binomial Theorem

Level 1 - Math I (Physics) topic page in Real Numbers.

The Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form \((a + b)^n\) where \(n\) is a non-negative integer.

Binomial Theorem

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

This expansion produces \(n + 1\) terms, with the coefficients given by binomial coefficients \(\binom{n}{k}\).

Expansion Pattern

For \((a + b)^n\):

Term \(k+1\) (for \(k = 0, 1, \ldots, n\)): \(\binom{n}{k} a^{n-k} b^k\)
Powers of \(a\) decrease from \(n\) to \(0\)
Powers of \(b\) increase from \(0\) to \(n\)

Examples

\((a + b)^2 = a^2 + 2ab + b^2\)

\((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

\((x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\)

Binomial Coefficients in Expansion

The coefficients \(\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}\) appear in the expansion of \((a + b)^n\). These can be found using Pascal's triangle or the factorial formula.

General Term

\[T_{k+1} = \binom{n}{k} a^{n-k} b^k \quad \text{for } k = 0, 1, \ldots, n\]

The Binomial Series

The binomial series extends the binomial theorem to cases where \(n\) is not a positive integer. For \(|x| < 1\):

Binomial Series

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

where for any real \(n\):

Generalized Binomial Coefficient

\[\binom{n}{k} = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}\]

Special Cases

Series for (1+x)^{-1}

\[(1 + x)^{-1} = 1 - x + x^2 - x^3 + \cdots\]

Series for (1+x)^{1/2}

\[(1 + x)^{1/2} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \cdots\]

Binomial Coefficients

Functions