How do you use the binomial series to expand #x^4/(1-3x)^3#?

1 Answer
Sep 17, 2017

Answer:

The expansion is #x^4/(1-3x)^3=x^4+9x^5+54x^6+270x^7+cdots#, and is valid for #|x|<1/3#.

Explanation:

The general binomial series expansion can be written as #(1+z)^p=1+pz+(p(p-1))/(2!)z^2+(p(p-1)(p-2))/(3!)z^3+cdots# for #|z|<1# (though the expansion is finite and works for all #z# if #p# is a non-negative integer).

For the given expression, we can write
#x^4/(1-3x)^3=x^4 * (1+(-3x))^(-3)# and use the expansion above with #z=-3x# and #p=-3#. This gives:

#x^4 * (1+(-3x))^(-3)=x^4(1-3(-3x)+((-3)*(-4))/(2!)(-3x)^2+#

#((-3) * (-4) * (-5))/(3!)(-3x)^3+cdots)#.

This simplifies to

#x^4(1+9x+6*9x^2-10*(-27)x^3+cdots)#

#=x^4+9x^5+54x^6+270x^7+cdots#, and is valid for #|x|<1/3# (which is equivalent to #|z|=|-3x|<1#).