General Binomial Theorem/Examples/(1-x)^(-3)
Jump to navigation
Jump to search
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 |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Exercises $\text {III}$: $1 \ \text {(a)}$