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

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Answer 1 · 2017-02-26T08:29:35

The answer is #=1-2x-2x^2-4x^3-10x^4+....#

The binomial series is

#(1+y)^n=sum_(k=0)^(oo)((n),(k))y^k#

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

Here, we have

#y=-4x#

#n=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+.......#

#=1-2x-2x^2-4x^3-10x^4+....#