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

1 Answer
Nov 24, 2016

Answer:

The answer is #=1/2-x/16+(3x^2)/256-(5x^3)/2048+......#

Explanation:

The binomial theorem is
#(a+b)^n=((n),(0))a^n+((n),(1))a^(n-1)b+((n),(3))a^(n-2)b^2+((n),(4))a^(n-3)b^3+......#

Where #((n),(p))=(n!)/((n-p)!p!)#

Let's rewrite #1/(sqrt(4+x))#

as #1/(2sqrt(1+x/4))=1/2(1+x/4)^(-1/2)#

Now we can use the binomial theorem,

#1/2(1+x/4)^(-1/2)#

#=1/2(1-1/2*x/4+1/2*3/2*1/2*x^2/16-1/2*3/2*5/2*1/6*x^3/64)#

#=1/2-x/16+(3x^2)/256-(5x^3)/2048+......#