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

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Answer 1 · 2016-06-25T16:26:04

$(1+2x)^(-3)=sum(-1)^n2^(n-1)(n+1)(n+2)x^n, n=0, 1, 2, 3,...$

Use #(1+X)^(-3)=1-3x+(3)(4)/(2!)X^2-...+(-1)^n((3)(4)(5)...(n+1)

(n+2))/(n!)X^n+..., |X|<1#

$=1-3x+((3)(4))/2X^2-...+(-1)^n((n+1)(n+2))/2)X^n+..., |X|<1$

Here, X = 2x.

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