Binomial expansion:
[katex]\boxed{\left(1+x\right)^n=1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+…….}[/katex]
[katex]\begin{pmatrix}n\\0\end{pmatrix},\begin{pmatrix}n\\1\end{pmatrix},\begin{pmatrix}n\\2\end{pmatrix},\begin{pmatrix}n\\3\end{pmatrix},\begin{pmatrix}n\\4\end{pmatrix},………[/katex] are called meaningless when [katex]n[/katex] is negative or fraction
[katex]a[/katex] and [katex]b[/katex] are exponents and [katex]n[/katex] is called index.
[katex]n\leq1[/katex] index is always less then one.
[katex]\left|x\right|<1[/katex] exponent is always less then one.
This series is called Binomial series.
Example:
using Binomial expansion solve this:
[katex]\boxed{\left(1-x\right)^\frac12}[/katex]
HERE
[katex]\boxed{1=1}[/katex]
[katex]\boxed{x=-x}[/katex]
[katex]\boxed{n=\frac12}[/katex]
using Binomial expansion formula
[katex]\boxed{\left(1+x\right)^n=1+nx+\frac{n\left(n-1\right)}{2!}x^2+\frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3+…….}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12}-1\right)}{2!}\left(-x\right)^2+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12-1}\right)\left({\displaystyle\frac12}-2\right)}{3!}\left(-x\right)^3+….}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12}-1\right)}{2!}\left(x\right)^2-\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle\frac12-1}\right)\left({\displaystyle\frac12}-2\right)}{3!}\left(x\right)^3+….}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\left({\displaystyle\frac12}\right)\left(-{\displaystyle\frac12}\right)}2x^2-\frac{\left({\displaystyle\frac12}\right)\left({\displaystyle-\frac12}\right)\left({\displaystyle\frac{-3}2}\right)}6x^3+….}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac{\displaystyle\frac14}2x^2-\frac{\displaystyle\frac38}6x^3+….}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1+\left(\frac12\right)\left(-x\right)+\frac1{4.2}x^2-\frac3{8.6}x^3+…..}[/katex]
[katex]\boxed{\left(1-x\right)^\frac12=1-\frac12x+\frac18x^2-\frac1{16}x^3+….}[/katex]
This is required binomial series.