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

Oct 23, 2016

the binomial expansion is $1 - 4 x + 12 {x}^{2} - 32 {x}^{3} + 80 {x}^{4} - 192 {x}^{5} + \ldots .$

#### Explanation:

We must write in the form${\left(1 + b\right)}^{n}$
The development of ${\left(1 + b\right)}^{n} = 1 + \frac{n}{1} \left(b\right) + \frac{n \left(n - 1\right)}{1 \cdot 2} {b}^{2} \cdot \ldots \ldots$
n can be positive or negative

So we apply this and we get

$\frac{1}{1 + 2 x} ^ 2 = {\left(1 + 2 x\right)}^{- 2} = 1 + \frac{- 2}{1} \left(2 x\right) + \frac{- 2 \cdot - 3}{1 \cdot 2} {\left(2 x\right)}^{2} + \frac{- 2 \cdot - 3 \cdot - 4}{1 \cdot 2 \cdot 3} {\left(2 x\right)}^{3} + \frac{- 2 \cdot - 3 \cdot - 4 \cdot - 5}{1 \cdot 2 \cdot 3 \cdot 4} {\left(2 x\right)}^{4} + \frac{- 2 \cdot - 3 \cdot - 4 \cdot - 5 \cdot - 6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} {\left(2 x\right)}^{5} + \ldots$
$= 1 - 4 x + 12 {x}^{2} - 32 {x}^{3} + 80 {x}^{4} - 192 {x}^{5} + \ldots \ldots .$