Euler's Continued Fraction in Lisp

In 1748, Leonhard Euler published a formula describing an identity that connected and generalized an infinite series and infinite continued fraction.

Euler's Equation

Euler's number, e, the constant 2.71828, is the base of the natural logarithms. Given n approaching infinity, Euler's number is the limit of:

\begin{align*}\displaystyle{\displaylines{(1 + 1/n)n}}\end{align*}

It's used frequently abroad across the sciences. It can also be elegantly expressed as an infinite series, like so:

\begin{align*} {\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .} \end{align*}

Separately, the imaginary unit i, \({\displaystyle {\sqrt {-i}}}\), represents the imaginary solution to the quadratic equation, x² + 1 = 0. The value can also be used to extend real numbers to complex numbers.

And π is pi, the irrational number we all know and love, roughly approximate to 3.14159, representing the ratio of the circle's circumference to its diameter.

While it isn't absolutely understood, we can join the three numbers in a seemingly bizarre proof that just works.

\( {\displaystyle e^{i\pi }=-1} \)