Here it is:
$latex \pi=16\sum_{m\geq 1}{(-1)^{m+1}\frac{m^2}{4m^2-1}}.$
Yes, it’s a divergent series, but I’m sure Euler would like it even more. (Actually, the probability that this formula is not somewhere in his works, or in Ramanujan’s, is close to zero, though I came upon it fairly accidentally today — maybe I’ll explain how it came about naturally at some later time).
Amusingly, both Pari/GP (numerically, using sumalt) and Maple (symbolically, after setting _EnvFormal:=true;) can confirm the “formula” as-is… (I didn’t try with Mathematica).
