Formulas

The most beautiful formulas:

Gauss's integral of

$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{2 \pi} $

Ramanujan sum:

$ \sum_{k=1}^{\infty} k = - \frac{1}{12} $

Zeta-regularized product:

$ \prod_{k=1}^{\infty} k = \sqrt{2 \pi} $

Euler formula:

$ e^{i \pi} + 1 = 0 $

Archimedes' Recurrence Formula:

$ a_{2n} = \frac{2 a_n b_n}{a_n + b_n} $

$ b_{2n} = \sqrt{a_{2n} b_n} $