The Lumber Room

"Consign them to dust and damp by way of preserving them"

(1+ix/n)^n

[Posting some images here for possible future reuse.]

$\displaystyle \lim_{n\to\infty}\left(1 + \frac{ix}{n}\right)^n = \cos x + i \sin x$

A non-rigorous argument: when $n$ is large enough so that $x/n$ is small, $(1 + ix/n)$ is roughly (hand-waving) the point on the unit circle at arc length (and hence angle) $x/n$:

So multiplication by $(1+ix/n)$ roughly corresponds to rotation by angle $x/n$. Multiplication by $(1+ix/n)^n$, which is multiplication by $(1+ix/n)$ n times, roughly corresponds to rotation by angle $n(x/n) = x$. As $n \to \infty$, “roughly” becomes exact.

Animation for $x = 1$:

Image generated from Python-generated SVG files; code available if anyone wants.

In particular, once one accepts the fact/definition that $\lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n = e^x$ (for instance, show that the function $f(x) = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n$ satisfies $f(x+y) = f(x)f(y)$), it is true that $e^{i\pi}$ is a rotation by π, that is,

$\displaystyle e^{i\pi} = -1$