How do you use the binomial series to expand #1/(1+2x)^3#?

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Answer 1 · 2015-12-17T04:47:22

#frac{1}{(1+2x)^3} = sum_{k=0}^oo (-1)^k (k+1)(k+2) 2^{k-1} x^k#

#-1/2<\x<1/2#

#(1 + x)^n = 1 + nx + frac{n(n-1)}{2!} x^2 + frac{n(n-1)(n-2)}{3!} x^3 + ...#

#frac{1}{(1+2x)^3} = (1+2x)^{-3}#

#= 1 + (-3)(2x) + frac{(-3)(-4)}{2}(2x)^2 + frac{(-3)(-4)(-5)}{(3)(2)}(2x)^3 + ...#

#= 1 - (3)(2x) + frac{(3)(4)}{2}(2)^2x^2 - frac{(3)(4)(5)}{(3)(2)}(2)^3x^3 + ...#

#= sum_{k=0}^oo (-1)^k frac{(k+1)(k+2)}{2} 2^k x^k #

#= sum_{k=0}^oo (-1)^k (k+1)(k+2) 2^{k-1} x^k #