jgk@osc.COM (Joe Keane) (05/22/91)
I found a nice pi series:
pi/2 = sum(k=0,1,2,...) (-1)^k*(6*k+5) * k!*(2*k-1)!!/(4*k+3)!!
For n odd we have n!! = 1*3*5*...*(n-4)*(n-2)*n, and note that -1!! = 1.
It seems like it'd be good for computing pi. The limiting ratio between terms
is 8, so you get almost one decimal place per term. And unlike arctangents,
there's only one series to sum. (Yes i know there are asymptotically faster
methods.) Of course when you're computing it, you should accumulate the
factorial part, multiplying by (k*(2*k-1))/((4*k+1)*(4*k+3)) each time.
It looks like an upside-down arcsine series. Has anyone seen it before? Is
it a special case of something? It's a hypergeometric series, but that hasn't
helped. Is there an easy way to prove that it works? I derived it, but it's
not pretty. I leave that as an exercise for the interested reader.
In case you want to see it `in action':
3.33333333333333333333
-20952380952380952381
+1962481962481962482
-204240204240204240
+22323737494016132
-2510165385384123
+287513546363997
-33362148669221
+3908690532122
-461352128473
+54776611387
-6534942378
+782738985
-94069565
+11337727
-1369867
+165871
-20123
+2445
-298
+36
-4
+1
======================
3.14159265358979323845
The last digit is wrong due to rounding error.
--
Joe Keane, amateur mathematician
jgk@osc.com (...!uunet!stratus!osc!jgk)