How do you use the binomial series to expand #f(x)= 1/(sqrt(1+x^3))#?

1 Answer
Steve M
Aug 13, 2017

# f(x) = 1 -1/2 x^3 + 3/8x^6 - 5/16 x^9...#

Explanation:

The binomial series tell 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:

# f(x) =1/sqrt(1+x^3) #
# " " =(1+x^3)^(-1/2) #

Then with #n=-1/2#, and replacing #x# in the definition with #x^3# we have:

# f(x) = 1+(-1/2)(x^3) + ( (-1/2)(-3/2) )/2(x^3)^2 + ( (-1/2)(-3/2)(-5/2))/6 (x^3)^3 + ...#

# " " = 1 -1/2 x^3 + (3/4)/2x^6 - (15/8)/6 x^9 + ...#

# " " = 1 -1/2 x^3 + 3/8x^6 - 5/16 x^9...#

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