Expressions for Pi

What are the different types of expressions of Pi?

1. As a series of rational numbers. Progressing ahead in a logical way.  pi/4 =1 -1/3+1/5-1/7+1/9 . . . . These are slow. A value of pi to 1000 places may take a day, let's say in a software. These are of complex nature of our experience of numbers. You may recall that -1 is i squared. Also possibly the 90 degree 1/2,1/4, 1/6 etc. are eaten up. Or you may think when 1 and 3 are there why should I think of 2. It is intended. A complex thought !!
2. As a series of factorials. pi/2 = (n!)/(2n+1)!!( Double factorial is different. n!! is not same as (n!)! Double factorial is multiplying the even numbers or the odd numbers as the case maybe.)  This is fast and efficient for pi calculations. A value of pi to 1000 places may take 5 minutes in software.
3. As s(n).(An+B/C^n) series. This was pioneered by Ramanujan. And the fastest and most efficient. I am not clear here. Ramanujan was a genius here and his notes are confusing.

Why should I express pi?

• Because it is nature
• Because I am naturally affected by this.
• Because it suits my thinking