Binomial Theorem/General Binomial Theorem/Proof 3

Proof
The series is the Maclaurin series expansion of the function $f(x) = \paren {1 + x}^\alpha$. The derivatives of $f$ are:


 * $f^{\paren 0}(x) = \paren {1 + x}^\alpha$
 * $f^{\paren 1}(x) = \alpha\paren {1 + x}^{\alpha-1}$
 * $f^{\paren 2}(x) = \alpha\paren{\alpha-1}\paren {1 + x}^{\alpha-2}$
 * $\cdots$
 * $\begin{align}

f^{\paren n}(x) &= \alpha\paren{\alpha-1}\cdots\paren{\alpha-n+1}\paren {1 + x}^{\alpha-n} \\ &=\alpha^{\underline n}\paren {1 + x}^{\alpha-n} \end{align}$

Evaluated at $0$, we have


 * $\begin{align}

f^{\paren n}(0) &= \alpha^{\underline n}\paren {1 + 0}^{\alpha-n} \\ &= \alpha^{\underline n} \end{align}$

The Maclaurin series of $f$ is: