The binomial expansion when n is fractional or negative is a little different in form than when n is positive. We are not able to use the standard method of color(white)(0)^nC_r, since this is only valid for positive integers. Notice also that we do not have a finite number of terms. The number of terms required would have to be specified.
The form we will use is:
(1+x)^n-= 1+nx+(n(n-1))/(2!)x^2+(n(n-1)(n-2))/(3!)x^3+..........
We have (6+x)^-3, we need to manipulate this to get the required form:
1/6^3(1+x/6)^-3
1+(-3)(x/6)+((-3)(-4))/2(x/6)^2+((-3)(-4)(-5))/6(x/6)^3->
+((-3)(-4)(-5)(-6))/24(x/6)^4+((-3)(-4)(-5)(-6)(-7))/120(x/6)^5
1/216(1-1/2x+1/6x^2-5/108x^3+5/432x^4-7/2592x^5+...)
1/216-1/432x+1/1296x^2-5/23328x^3+5/93312x^4-7/559872x^5+...
These series are always infinite, unless |x|<1, or in this case |x/6|<1
