Infinite Expressions for Pi

and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for  to be

 .

 ,

and the fact that arctan(1) = /4 to obtain the series

 .

Unfortunately, this series converges to slowly to be useful, as it takes over 300 terms to obtain a 2 decimal place precision. To obtain 100 decimal places of , one would need to use at least 10^50 terms of this expansion!

 ,

and the fact that arctan(1/2) = /6 to obtain the series

 .

This arcsine series converges much faster than using the arctangent. (Actually, Newton used a slightly different expansion in his original text.) This expansion only needed 22 terms to obtain 16 decimal places for .