Category:Examples of Use of General Binomial Theorem

This category contains examples of use of General Binomial Theorem.

Let $\alpha \in \R$ be a real number.

Let $x \in \R$ be a real number such that $\size x < 1$.

Then:

$\ds \paren {1 + x}^\alpha$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n$

$\ds $

$=$

$\ds 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots$

where:

$\alpha^{\underline n}$ denotes the falling factorial

$\dbinom \alpha n$ denotes a binomial coefficient.

Pages in category "Examples of Use of General Binomial Theorem"

The following 5 pages are in this category, out of 5 total.