Recently, I was experimenting with formulas involving circles and convergent series and came across a new type of series which seems to converge to $\pi$ extremely fast. I have yet to see this anywhere so I was curious to see if it is completely unique or if it already exists. In the case that is unique, would it be safe for me to share it here and would it even be considered a significant find?
Edit: I have inserted my series below, feel free to check if it works
$$ \sum_{n=0}^{\infty } \frac{1}{2}\left ( 1000^{1000n} \right )!\left \{ \sin\left [ \frac{360}{\left ( 1000^{1000n} \right )!} \right ] \right \}=\pi $$
This has the form $$\sum_{n=0}^\infty\frac{m_n}2\left(\sin\frac{360^\circ}{m_n}\right)$$ where $m_n\to \infty$. But $$\lim_{m\to\infty}\frac m2\left(\sin\frac{360^\circ}m\right)=\pi$$ so your series diverges.
Also I am not aware of any method of numerically calculating $\sin x^\circ$ which doesn't require knowledge of $\pi$ to a similar accurracy.