W. Lang, Aug 07 2007 Rationals r(n)= A131445(n)/ A131446(n). r(n):=sum(b(k)/k!,n=1..n) with b(k):=A075874(k) (factorial expansion of Pi). A075874(k), k=1..30: [3, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9,...] r(n), n=1..25: [3, 3, 3, 25/8, 47/15, 2261/720, 15833/5040, 42223/13440, 42223/13440, 11400211/3628800, 1672031/532224, 136802537/43545600, 2173640311/691891200, 2173640311/691891200, 342348348983/108972864000, 5975534818613/1902071808000, 372475003693547/118562476032000, 21511925347007/6847458508800, 76431870757915873/24329020081766400, 56199904969055789/17888985354240000, 4223866541884824563/1344498478202880000, 1765576214507856667337/562000363888803840000, 111868465382040505093/35608838483312640000, 114658596518627868278849/36496964807837614080000, 1624330117347228133950361/517040334777699532800000] The values r(10^k), k=0..3, are (maple11; 10 digits): [3., 3.141592537, 3.141592654, 3.141592654] This should be compared with Pi (maple11; 10 digits) 3.141592654 . ##################################### e.o.f. ##################################