General Binomial Theorem/Examples/(1-x)^(-3)

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Example of Use of General Binomial Theorem

$\paren {1 - x}^{-3} = 1 + 3 x + 6 x^2 + 10 x^3 + \cdots$


Proof

\(\ds \paren {1 - x}^{-3}\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-3}^{\underline n} } {n!} \paren {-x}^n\) General Binomial Theorem
\(\ds \) \(=\) \(\ds 1 + \paren {-3} \paren {-x} + \dfrac {\paren {-3} \paren {-4} } {2!} \paren {-x}^2 + \dfrac {\paren {-3} \paren {-4} \paren {-5} } {3!} \paren {-x}^3 + \cdots\) expanding term by term
\(\ds \) \(=\) \(\ds 1 + 3 x + \dfrac {12} 2 x^2 + \dfrac {60} 6 x^3 + \cdots\) simplifying
\(\ds \) \(=\) \(\ds 1 + 3 x + 6 x^2 + 10 x^3 + \cdots\) simplifying

$\blacksquare$


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