Using Taylor Series to approximate pi without using arctan?

Question

Whenever I search for ways to approximate pi using Taylor/MacLaurin Series, the example that I always see utilizes the fact that $\tan{\frac{\pi}{4}=1}$. However, I vaguely remember coming across a document that suggested this wasn't the best way due to the very slow convergence. I believe they provided an alternate, quite simple example that didn't utilize arctan at all and also converged much faster. However, I can no longer find the document. Is there such an example? I'm looking for the following properties:

1. Simple equation that uses pi.
2. Doesn't use arctan.
3. Converges quickly.

Thank you!