Binomial Theorem

Binomial Coefficient $\left(\genfrac{}{}{0ex}{}{n}{m}\right)=\frac{n!}{(n-m)!m!}$ The binomial theorem is one of the most important results in mathematics.

It builds on top of Pascal’s Identity (PI).

Binomial Theorem, Version 1 (BT1)

For all integers $n\ge 0$ and all real numbers $x$,

$(1+x{)}^n={\sum }_{m=0}^n(\genfrac{}{}{0ex}{}{n}{m})x^m$

Binomial Theorem, Version 2 (BT2)

For all integers $n\ge 0$ and all real/complex numbers $a$ and $b$,

$(a+b{)}^n={\sum }_{m=0}^n(\genfrac{}{}{0ex}{}{n}{m})a^{n-m}{b}^m$

Binomial Series

The Binomial Series is a generalization of the Binomial Theorem. It removes the restriction that $m$ need be an integer (and, to make it easier to use with functions, we set $a=1$). Specifically, it allows us to write

$(1+x{)}^a={\sum }_{k=0}^{\infty }(\genfrac{}{}{0ex}{}{a}{k})x^k$ In expanded form, $(1+x{)}^m=1+mx+m(m-1)\frac{x^2}{2!}+m(m-1)(m-2)\frac{x^3}{3!}+\cdots +m(m-1)(m-2)\cdots (m-n+1)\frac{x^n}{n!}+\cdots$

Furthermore, it has radius of convergence 1.

Related

• Negative Binomial Theorem