Binomial Theorem/General Binomial Theorem/Proof 2

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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

From Sum over k of r-kt choose k by r over r-kt by z^k:

$\ds \sum_n \dbinom {\alpha - n t} k \dfrac \alpha {\alpha - n t} z^n = x^\alpha$

where:

$z = x^{t + 1} - x^t$
$x = 1$ for $z = 0$.

Setting $t = 0$:

\(\ds \sum_k \dbinom {\alpha - n \times 0} n \dfrac \alpha {\alpha - n \times 0} z^n\) \(=\) \(\ds x^\alpha\)
\(\ds \leadsto \ \ \) \(\ds \sum_n \dbinom \alpha n \dfrac \alpha \alpha z^n\) \(=\) \(\ds \paren {1 + z}^\alpha\)
\(\ds \leadsto \ \ \) \(\ds \sum_n \dbinom \alpha n z^n\) \(=\) \(\ds \paren {1 + z}^\alpha\)

$\blacksquare$


Historical Note

The General Binomial Theorem was first conceived by Isaac Newton during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe.

He announced the result formally, in letters to Henry Oldenburg on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate level of rigor had not been recognised).

Leonhard Paul Euler made an incomplete attempt in $1774$, but the full proof had to wait for Carl Friedrich Gauss to provide it in $1812$.

This was, in fact, the first time anything about infinite summations was proved adequately.


Sources

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