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

Jan 5, 2016

$1 + \frac{1}{2} {x}^{2} + \frac{3}{8} {x}^{4} + \frac{15}{48} {x}^{6}$

#### Explanation:

Rewrite $\frac{1}{1 - {x}^{2}} ^ \left(\frac{1}{2}\right)$ as

${\left(1 - {x}^{2}\right)}^{- \frac{1}{2}}$

since there is a negative index the only formula that can be used is

 1 + na +( n(n - 1 ))/(2!) a^2 +( n(n- 1)(n - 2 ))/(3! )a^3 + ...

here $n = - \frac{1}{2} , a = - {x}^{2}$

≣ 1 +  (-1/2 )(- x^2 ) +( (- 1/2)(- 3/2))/(2!) (- x^2 )^2 + ((-1/2)(-3/2)(-5/2)) /(3!) (-x^2)^3

be very careful when expanding.

≣ 1 + 1/2 x^2 + 3/8 x^4 + 15/48 x^6