#(a+b)^n=a^n(1+b/a)^n=a^n(1+n(b/a)+(n(n-1)(b/a)^2)/(2!)+(n(n-1)(n-2)(b/a)^3)/(3!)+cdots)#
For #(4-3x^2)^(-1/2)#
We have #4^(-1/2)(1-(3x^2)/4)^(-1/2)=(1-(3x^2)/4)^(-1/2)/2#
#(1-(3x^2)/4)^(-1/2)=1+(-1/2)((-3x^2)/4)+((-1/2)(-3/2)((-3x^2)/4)^2)/(2!)+((-1/2)(-3/2)(-5/2)((-3x^2)/4)^3)/(3!)#
#color(white)((1-(3x^2)/4)^(-1/2))=1+(3x^2)/8+((3/4)(9x^4)/16)/2+((-15/8)(-27x^6)/64)/6#
#color(white)((1-(3x^2)/4)^(-1/2))=1+(3x^2)/8+((27x^4)/64)/2+((405x^6)/512)/6#
#color(white)((1-(3x^2)/4)^(-1/2))=1+(3x^2)/8+(27x^4)/128+(405x^6)/3072#
#color(white)((1-(3x^2)/4)^(-1/2))=1+(3x^2)/8+(27x^4)/128+(135x^6)/1024#
#1/2(1-(3x^2)/4)^(-1/2)=(1+(3x^2)/8+(27x^4)/128+(135x^6)/1024)/2=1/2+(3x^2)/16+(27x^4)/256+(135x^6)/2048#
