Binomial Theorem/General Binomial Theorem/Proof 3
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Theorem
Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
\(\ds \paren {1 + x}^\alpha\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots\) |
where:
- $\alpha^{\underline n}$ denotes the falling factorial
- $\dbinom \alpha n$ denotes a binomial coefficient.
Proof
The series is the Maclaurin series expansion of the function $\map f x = \paren {1 + x}^\alpha$.
The derivatives of $f$ are:
\(\ds \map {f^{\paren 0} } x\) | \(=\) | \(\ds \paren {1 + x}^\alpha\) | ||||||||||||
\(\ds \map {f^{\paren 1} } x\) | \(=\) | \(\ds \alpha \paren {1 + x}^{\alpha - 1}\) | ||||||||||||
\(\ds \map {f^{\paren 2} } x\) | \(=\) | \(\ds \alpha \paren {\alpha - 1} \paren {1 + x}^{\alpha - 2}\) | ||||||||||||
\(\ds \map {f^{\paren n} } x\) | \(=\) | \(\ds \alpha \paren {\alpha - 1} \cdots \paren {\alpha - n + 1} \paren {1 + x}^{\alpha - n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha^{\underline n} \paren {1 + x}^{\alpha - n}\) |
Evaluated at $x = 0$, we have:
\(\ds \map {f^{\paren 0} } x\) | \(=\) | \(\ds \alpha^{\underline n} \paren {1 + 0}^{\alpha - n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha^{\underline n}\) |
The Maclaurin series of $f$ is:
\(\ds \map f x)\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \alpha^{\underline n}\) | substituting derivative at $0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n\) | rearranging |
$\blacksquare$