In his second letter to Hardy, Ramanujan stated:

$ 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}$

Hardy could tell Ramanujan was a genius from this equation because he must have developed the theory of the zeta function, and its functional equation.

In this excellent note by Dror Bar-Natan, we start with the following equations:

$ 1 \times 2 \times 3 \times 4 \times \cdots = \sqrt{2 \pi}$ $ 1 + 1 + 1 + 1 + \cdots = -\frac{1}{2}$

and are asked to derive the Wallis' formula for $\pi$. This will use a couple of double factorial identities.

$ \frac{\pi}{2} $

$ = \frac{1}{4} \times 2 \pi $

$ = {2^{-\frac{1}{2}}}^4 \times {\infty !}^2 $

$ = \frac{{2^{-\infty}}^4 {\infty !}^4}{{\infty !}^2} $

$ = \frac{{\infty !!}^4}{{\infty !}^2} $

$ = \frac{2 \times 2 \times 2 \times 2 \times 4 \times 4 \times 4 \times 4 \times 6 \times 6 \times 6 \times 6 \cdots}{1 \times 2 \times 2 \times 3 \times 3 \times 4 \times 4 \times 5 \times 5 \times 6 \times 6 \times 7 \times 7 \cdots} $

$ = \frac{2}{1} \frac{2}{3} \frac{4}{3} \frac{4}{5} \frac{6}{5} \frac{6}{7} \cdots \qed $

What is infinity factorial (and why might we care)?

Srinivasa Ramanujan - G. H. Hardy

The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation - Terence Tao

On the Number of Primes Less Than a Given Magnitude - Bernhard Riemann

Double Factorial - Mathworld

Expository preaching, non-rigorous proofs

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