Estimating Pi

I tried estimating pi in an 
interesting manner. Without proof, just accept that 

pi^2/6 = SUM 1/n^2

(note: SUM, without any index markings, means sum
from n=1 to infinity)

So then we can do some algebraic manipulations as follows:

pi^2 = 6* SUM 1/n^2

pi = sqrt[ 6* SUM 1/n^2 ]

So then I wanted to figure out how many terms of the SUM I 
would need to approximate pi to the hundreds place; i.e.,
what partial sum would give me 3.14. I worked it out on 
Mathematica and got that N=598 will do it. So,
 
           N
 sqrt[6 * SUM 1/n^2 ]
           1

3.14 <-> N=598
3.141 <-> N=1598
3.1415 <-> N=9779
3.14159 <-> N>80000 (the computer was taking forever to compute
                      it so I just stopped)

The last I computed was N=80000 and I got 3.14158 so to get 
the 9 out there, we need N>80000.

So, as you can see, the convergence is very very slow.


...pretty cool stuff...


jmg
4/27/05