what is the equation of pi?
2 answers
Hi - there are any number of them, but they're all infinite series: if you go to Wikipedia and look up "pi", you'll find at least fifteen of them in the "Computation in the Computer Age" section of that entry.
If you are looking to calculate π numerically, there are many series that will eventually converge to π, none of which will yield the exact value with a finite number of terms.
For example, the famous Gregory-Leibniz series will converge rather slowly (requires 10n terms for n accurate digits):
π/4 = 1-1/3+1/5-1/7+1/9-...
The Borweins and Dilcher have improved on the convergence of the above series, giving 136 digits with 100 terms:
http://mathpath.net/leibniz.htm
One of my favourites is the one from T.J.Bromwich(1926):
π/2=1+(2/3)/2+(2*4/(3*5))/2²+(2*4*6/(3*5*7))/2³+....
because it converges relatively rapidly, and each term can be derived from the previous.
For example, the famous Gregory-Leibniz series will converge rather slowly (requires 10n terms for n accurate digits):
π/4 = 1-1/3+1/5-1/7+1/9-...
The Borweins and Dilcher have improved on the convergence of the above series, giving 136 digits with 100 terms:
http://mathpath.net/leibniz.htm
One of my favourites is the one from T.J.Bromwich(1926):
π/2=1+(2/3)/2+(2*4/(3*5))/2²+(2*4*6/(3*5*7))/2³+....
because it converges relatively rapidly, and each term can be derived from the previous.
1 answer