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

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#### Explanation

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#### Explanation:

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May 1, 2018

${\left(1 + x\right)}^{\frac{1}{2}} = 1 + \frac{1}{2} x - \frac{1}{8} {x}^{2} + \frac{1}{16} {x}^{3} - \frac{5}{128} {x}^{4} + \ldots \ldots .$

#### Explanation:

According to binomial series (1+a)^n=1+na+(n(n-1))/(2!)a^2+(n(n-1)(n-2))/(3!)a^3+(n(n-1)(n-2)(n-3))/(4!)a^4+.......

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

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

or ${\left(1 + x\right)}^{\frac{1}{2}} = 1 + \frac{1}{2} x - \frac{1}{8} {x}^{2} + \frac{1}{16} {x}^{3} - \frac{5}{128} {x}^{4} + \ldots \ldots .$

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