数列の極限による自然対数の底の定義

$ e^x = \lim_{n\to\infty} \left( 1 + \frac{x}{n} \right)^n

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\begin{aligned}

S_n

& \coloneqq \left( 1 + \frac{x}{n} \right)^n\\

&= \binom{n}{0} + \binom{n}{1}\left(\frac{x}{n}\right) + \binom{n}{2}\left(\frac{x}{n}\right)^2 + \binom{n}{3}\left(\frac{x}{n}\right)^3 + \cdots\\

&= \binom{n}{0} + \frac{1}{n}\binom{n}{1}x + \frac{1}{n^2}\binom{n}{2}x^2 + \frac{1}{n^3}\binom{n}{3}x^3 + \cdots\\

&= 1 + \frac{1}{n}nx + \frac{1}{n^2}\frac{1}{2!}n(n-1)x^2 + \frac{1}{n^3}\frac{1}{3!}n(n-1)(n-2)x^3 + \cdots\\

&= 1 + x + \frac{1}{2}\left(1-\frac{1}{n}\right)x^2 + \frac{1}{6}\left(1-\frac{3}{n}-\frac{2}{n^2}\right)x^3 + \cdots\\

\lim_{n\to\infty}S_n

&= 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3 + \cdots\\

&= e^x

\end{aligned}