The following formula for π was discovered by Ramanujan:
1π=2√29801∞∑k=0(4k)!(1103+26390k)(k!)43964kDoes anyone know how it works, or what the motivation for it is?
Answer
Here’s an easy introduction to the basics, “Pi Formulas and the Monster Group“.
http://sites.google.com/site/tpiezas/0013
Update: Just to make this more intriguing, define the fundamental unit U29=5+√292 and fundamental solutions to Pell equations,
(U29)3=70+13√29,thus702−29⋅132=−1
(U29)6=9801+1820√29,thus98012−29⋅18202=1
26((U29)6+(U29)−6)2=3964
then we can see those integers all over the formula as,
1π=2√29801∞∑k=0(4k)!k!429⋅70⋅13k+1103(3964)k
See also this MO post.
Attribution
Source : Link , Question Author : Nick Alger , Answer Author : Community