Binomial Theorem in one variable

The expansion of $n$-th power of a binomial in one variable is called the binomial theorem in one variable.

Introduction

Let $x$ be a real number $x \,∈\, R$ and a constant $n$ belongs to real numbers $n \,∈\, R$. The expansion of binomial theorem in one is written in the following forms.

Sum basis Binomial Theorem

$(1+x)^n$ $\,=\,$ $1$ $+$ $nx$ $+$ $\dfrac{n(n-1)}{2!}x^2$ $+$ $\dfrac{n(n-1)(n-2)}{3!}x^3$ $+$ $\cdots$

$(1+x)^{-n}$ $\,=\,$ $1$ $-$ $nx$ $+$ $\dfrac{n(n-1)}{2!}x^2$ $+$ $\dfrac{n(n-1)(n-2)}{3!}x^3$ $+$ $\cdots$

Difference basis Binomial Theorem

$(1-x)^n$ $\,=\,$ $1$ $-$ $nx$ $+$ $\dfrac{n(n-1)}{2!}x^2$ $-$ $\dfrac{n(n-1)(n-2)}{3!}x^3$ $+$ $\cdots$

$(1-x)^{-n}$ $\,=\,$ $1$ $+$ $nx$ $+$ $\dfrac{n(n+1)}{2!}x^2$ $+$ $\dfrac{n(n+1)(n+2)}{3!}x^3$ $+$ $\cdots$