\(\pi\) as the equivalence class of all partial sums convergent to it

$$f(k)\equiv\frac{k!}{(2k+1)!!}\equiv \frac{k!\,(2k)!\,(25k-3)}{(3k)!\,2^{k}}\equiv \sqrt{2}\frac{(-1)^{(n^2-n)/2}}{2n+1}\equiv  \frac{2(-1)^{n}}{2n+1}$$