How do you use the binomial series to expand # 1/(2+x)^3#?

1 Answer
Mar 23, 2018

Answer:

The answer is #=1/8-3/16x+3/16x^2-5/32x^3+o(x^3)#

Explanation:

The binomial series is

#(a+b)^n=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+.....#

Here,

#a=2#

#b=x#

and

#n=-3#

Therefore,

#1/(2+x)^3=(2+x)^(-3)#

#=2^(-3)+(-3)*(2^(-4))*x+(-3*-4)/(2)*2^(-5)x^2+(-3*-4*-5)/(6)2^(-6)x^3+....#

#=1/8-3/16x+3/16x^2-5/32x^3+o(x^3)#