Find a series expansion for? : # (1-3x)^(2/3) #
1 Answer
Oct 11, 2017
# (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 + ... #
