Rationals r(n):=A130411(n)/A130412(n):

r(n):=  r(n):=3*sum(((-1)^(j+1))/(j*(j+1)*(2*j+1)),j=1..n), n>=1.

They result from the partial sums 

s(n):=sum(((-1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..n)

which have the limit  (Pi-3)/4, approximately 0.0353981635,

after multiplication with 3*4=12.

The series for Pi/4 = 3/4 + sum(((-1)^(j+1))/(2*j*(2*j+1)*(2*j+2)),j=1..infinity) is attributed to

Nilakantha (c.1450-c.1550), Tantrasangraha (c.1500). See the quoted R. Roy reference, eq.(13).  


Therefore r(n) tends to 3*(Pi-3), approximately 0.424777962, for n->infinity. 


r(n), n=1..30:

[1/2, 2/5, 61/140, 44/105, 989/2310, 6346/15015, 51197/120120, 36056/85085, 4127401/9699690, 2057402/4849845, 189721879/446185740, 236723324/557732175, 1422382919/3346393050, 20600649518/48522699225, 10227626700773/24067258815600, 638723926928/1504203675975, 1278290544991/3008407351950, 23635180313246/55655536011075, 94585786464329/222622144044300, 969106771716436/2281876976454075, 83372817133541471/196241419975050450, 41673480936996358/98120709987525225, 15673494950136175183/36893386955309484600, 13711014028962429224/32281713585895799025, 27427870902012803389/64563427171791598050, 726700622862373923146/1710930820052477348325, 2907296266577937965989/6843723280209909393300, 726713168872896957316/1710930820052477348325, 85763953449898754600173/201889836766192327102350, 2615474951452934285745434/6157640021368865976621675]



Some values for r(10^k), k=0..2, are:

[.5000000000, .4242201555, .4247772329]  

which should be compared with 

0.424777962, approximately  3*(Pi-3).



s(n) for n=1..30:

[1/24, 1/30, 61/1680, 11/315, 989/27720, 3173/90090, 51197/1441440, 9014/255255, 4127401/116396280, 1028701/29099070, 189721879/5354228880, 59180831/1673196525, 1422382919/40156716600, 10300324759/291136195350, 10227626700773/288807105787200, 159680981732/4512611027925, 1278290544991/36100888223400, 11817590156623/333933216066450, 94585786464329/2671465728531600, 242276692929109/6845630929362225, 83372817133541471/2354897039700605400, 20836740468498179/588724259925151350, 15673494950136175183/442720643463713815200, 3427753507240607306/96845140757687397075, 27427870902012803389/774761126061499176600, 363350311431186961573/10265584920314864089950, 2907296266577937965989/82124679362518912719600, 181678292218224239329/5132792460157432044975, 85763953449898754600173/2422678041194307925228200, 1307737475726467142872717/36945840128213195859730050]

Some values for s(10^k), k=0..2, are:

[0.04166666667, 0.03535167962, 0.03539810274] 

which should be compared with 

0.0353981635, approximately (Pi-3)/4,


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