Question

Find a series expansion for? : `(1-3x)^(2/3)`

Answer 1

`(1-3x)^(2/3) = 1 - 2x - x^2 + 4/3x^3 + ...`

Explanation:

We seek an expansion of :

`(1-3x)^(2/3)`

The Binomial Series tells us that:

`(1+X)^n = 1+nX + (n(n-1))/(2!)X^2 (n(n-1)(n-2))/(3!)X^3 + ...`

And so for the given function we can expand using the Binomial Series as follows::

`f(x) = 1+(2/3)(-3x) + ((2/3)(-1/3))/(2!)(-3x)^2 + ((2/3)(-1/3)(-4/3))/(3!)(-3x)^3 + ...`

`\ \ \ \ \ \ = 1 - 2x - (2)/(2)x^2 + (8)/6x^3 + ...`

`\ \ \ \ \ \ = 1 - 2x - x^2 + 4/3x^3 + ...`