How do you use the binomial series to expand #4*sqrt(1+x)#?

1 Answer
Oct 29, 2016

Answer:

The answer is #=4(1+x/2-x^2/8+x^3/16+....)#

Explanation:

Use #(a+b)^n=a^n+(""_1^n)a^(n-1)b+(""_2^n)a^(n-2)b^2.....#
#=a^n+na^(n-1)b+((n)(n-1))/(1*2)a^(n-2)b^2+((n)(n-1)(n-2))/(1*2*3).a^(n-3)b^3....#
If we use the binomial development, we get
#4(1+x)^(1/2)=4(1+x/2+((1/2)(_1/2))/(1*2)x^2+((1/2)(-1/2)(-3/2))/(1*2*3)x^3+....)#

#=4(1+x/2-x^2/8+x^3/16+....)#