# How do you use the binomial series to expand (1-4x)^(1/2)?

Feb 26, 2017

The answer is $= 1 - 2 x - 2 {x}^{2} - 4 {x}^{3} - 10 {x}^{4} + \ldots .$

#### Explanation:

The binomial series is

${\left(1 + y\right)}^{n} = {\sum}_{k = 0}^{\infty} \left(\begin{matrix}n \\ k\end{matrix}\right) {y}^{k}$

=1+ny+(n(n-1))/(2!)y^2+(n(n-1)(n-2))/(3!)y^3+.....

Here, we have

$y = - 4 x$

$n = \frac{1}{2}$

Therefore,

(1-4x)^(1/2)=1+(1/2)(-4x)+((1/2)(-1/2))/(2!)(-4x)^2+((1/2)(-1/2)(-3/2))/(3!)(-4x)^3+((1/2)(-1/2)(-3/2)(-5/2))/(4!)(-4x)^4+.......

$= 1 - 2 x - 2 {x}^{2} - 4 {x}^{3} - 10 {x}^{4} + \ldots .$