# Binomial Theorem

$$\require{AMSsymbols} \def\lfz#1{\overset{\Large#1}{\,\circ\kern-6mu-\kern-7mu-\kern-7mu-\kern-6mu\bullet\,}} \def\lfzraised#1{\raise{10mu}{#1}} \def\laplace{\lfz{\mathscr{L}}} \def\fourier{\lfz{\mathcal{F}}} \def\ztransform{\lfz{\mathcal{Z}}} \require{cancel} \newcommand\ccancel[2][black] {\color{#1}{\cancel{\color{black}{#2}}}}$$Consider
$$f(x)=(a+x)^r\label{eq:axr}$$

Recall the MacLaurin Series

\begin{align} f(x)&=\sum _{k=0}^{\infty }{\frac {f^{(k)}(0)}{k!}}\,x^{k}\nonumber \end{align}\nonumber

The $$k$$th derivative of equation $$\eqref{eq:axr}$$
$$f^{(k)}(x)=r\,(r-1)\cdots(r-k+1)(a+x)^{r-k}$$

Substitute $$x=0$$ to find the derivatives at $$0$$
$$f^{(k)}(0)=r\,(r-1)\cdots(r-k+1)\,a^{r-k}$$

Apply the MacLaurin Series to equation $$\eqref{eq:axr}$$
\begin{align} f(x)=(a+x)^r&=\sum _{k=0}^{\infty }\frac{r\,(r-1)\cdots(r-k+1)}{k!}\,a^{r-k}\,x^k \end{align}

Isaac Newton generalized binomial theorem for $$r\in\mathbb{C}$$
$$\shaded{(a+x)^r=\sum_{k=0}^{\infty}{r \choose k}\,a^{r-k}\,x^k,\quad\text{where }{r \choose k}=\frac{r\,(r-1)\cdots(r-k+1)}{k!}}\label{eq:newton}$$

The binomial coefficient $${r \choose k}$$
\begin{align}\frac{(r)_k}{k!}&=\frac{r(r-1)(r-2)\cdots (r-k+1)}{k(k-1)(k-2)\cdots1}\nonumber\\ &=\prod _{i=1}^{k}\frac{(r-(i-1))}{i}=\prod _{i=0}^{k-1}\frac{r-i}{i} \end{align}

The series converges for $$r\geq0\land r\in\mathbb{N}$$, or for $$|x|\lt|a|$$
\begin{align} (a+x)^r&=\sum_{k=0}^{\infty}{r \choose k}\,a^{r-k}\,x^k\nonumber\\ &=a^r+r\,a^{r-1}\,x+\frac{r(r-1)}{2!}\,a^{r-2}\,x^2+\frac{r(r-1)(r-2)}{3!}\,a^{r-3}\,x^3+\cdots \end{align}

## Binomial series

Consider equation $$\eqref{eq:newton}$$ for $$a=1$$, gives the Binomial series
$$\shaded{(1+x)^r=\sum_{k=0}^{\infty}{r \choose k}\,x^k,\quad\text{where }{r \choose k}=\frac{r\,(r-1)\cdots(r-k+1)}{k!}}$$

This series converges when

• $$|x|\lt1$$, converges absolutely for any complex number $$r$$.
• $$|x|\gt1$$, converges only when $$r$$ is a non-negative integer, what makes the series finite.

## Special cases

1) where $$a=1$$, converges for $$|x|\lt1$$
\begin{align} (1+x)^{r}&=\sum_{k=0}^{\infty}\frac{(r)_k}{k!}\,x^k\nonumber\\ &=1+r\,x+\frac{r(r-1)}{2!}\,x^2+\frac{r(r-1)(r-2)}{3!}\,x^3+\cdots \end{align}

2) the negative binomial series, converges for $$|x|\lt1$$

Apply the Negated Upper Index of Binomial Coefficient identity $${r \choose k}=(-1)^k{k-r-1 \choose k}$$
\begin{align} (a+x)^r&=\sum_{k=0}^{\infty}{r \choose k}\,a^{r-k}\,x^k\nonumber\\ &=\sum_{k=0}^{\infty}{k-r-1 \choose k}(-1)^k\,\,a^{r-k}\,x^k \end{align}

Substitute $$x\to -x$$ and $$m\to -m$$
\begin{align} (a-x)^{-r}&=\sum_{k=0}^{\infty}{k+r-1 \choose k}(-1)^k\,\,a^{-r-k}\,(-x)^k\nonumber\\ &=\sum_{k=0}^{\infty}{k+r-1 \choose k}\cancel{(-1)^k}\,\,a^{-r-k}\,\cancel{(-1)^k}\,x^k\nonumber\\ &=\sum_{k=0}^{\infty}{k+r-1 \choose k}\,a^{-r-k}\,x^k \end{align}

For $$a=1$$
\begin{align} (1-x)^{-r}&=\sum_{k=0}^{\infty}{k+r-1 \choose k}\,x^k\nonumber\\ \end{align}

—-old:
\begin{align} (1-x)^{-r}&=\sum_{k=0}^{\infty}{-r \choose k}\,(-x)^k=\sum_{k=0}^{\infty}\frac{(-r)_k}{k!}\,(-x)^k\nonumber\\[5mu] &=1+r\,x+\frac{r(r+1)}{2!}\,x^2+\frac{r(r+1)(r+2)}{3!}\,x^3+\cdots \end{align}

3) where $$r=-1$$ the geometric series, converges for $$|x|\lt1$$
\begin{align} (1+x)^{-1}&=\sum_{k=0}^{\infty}\frac{(-1)_k}{k!}\,x^k\nonumber\\ &=1-x+x^2-x^3+\cdots \end{align}

## Integer exponents

1) when $$n=r$$ is a positive integer, the binomial coefficients for $$k\gt r$$ are zero, and the series terminates at $$n$$
$$(a+x)^{n}=\sum _{k=0}^{n}{n \choose k}\,a^{n-k}\,x^{k}$$ where $${n \choose k}=\frac{n!}{(n-k)!k!}$$ for nonnegative $$n$$ and $$k$$, and $$0\leq k\lt n$$

2) when the exponent $$-n=-r$$ is a negative integer, where $$|x|\lt a$$
\begin{align} (a+x)^{-n}&=\sum _{k=0}^{\infty}{-n \choose k}\,a^{-(n+k)}\,x^{k}\nonumber\\[5mu] &=\sum _{k=0}^{\infty}(-1)^k{n+k-1 \choose k}\,a^{-(n+k)}\,x^{k} \end{align}

For $$a=1$$, this simplifies to
\begin{align} (1+x)^{-n} &=\sum _{k=0}^{\infty}{-n \choose k}\,x^{k}\nonumber\\[5mu] &=1-nx+\frac{n(n+1)}{2!}x^2+\frac{n(n+1)(n+2)}{3!}x^3+\cdots \end{align}

Embedded software developer
Passionately curious and stubbornly persistent. Enjoys to inspire and consult with others to exchange the poetry of logical ideas.

This site uses Akismet to reduce spam. Learn how your comment data is processed.