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

1 Answer
Sep 17, 2017

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).