More commaents on the rational numbers A096952(n)/ A096953(n).

The upper bounds R(2*n), n=0..15, for 

         ln(2)-2*sum(((1/3)^(2*k-1))/(2*k-1),k=1..n) are:


[5/6, 35/648, 55/7776, 2315/1959552, 20195/90699264, 179195/3990767616, 

1602515/169789022208, 2876335/1410554953728, 129271235/287753210560512, 

1162785755/11577835060199424, 1494635765/65810851921133568, 

94151567435/18163795130232864768, 33892886555/28430288029929701376, 

7625731702715/27634239965091669737472, 

68630914235795/1068523945316877896515584, 

617675543767595/41119749068056404569358336]


Their values are approximately (maple9 10 digits)

[.8333333333, 0.5401234568e-1, 0.7073045267e-2, 0.1181392482e-2, 

0.2226589182e-3, 0.4490238902e-4, 0.9438272152e-5, 0.2039151323e-5, 

0.4492434150e-6, 0.1004320539e-6, 0.2271108368e-7, 0.5183474420e-8, 

0.1192140105e-8, 0.2759522865e-9, 0.6422964552e-10, 0.1502138407e-10]

where e_n means 10^(-n). 

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ln(2)-2*sum(((1/3)^(2*k-1))/(2*k-1),k=1..n) gives for n=0..10 (if n=0 

one puts the sum to zero) the approximate numbers: (maple9 15 digits)


[.693147180559945, 0.26480513893278e-1, 0.1789155868587e-2, 

0.143065333608e-3, 0.12423227657e-4, 0.1133169118e-5, 0.106800160e-6, 

0.10303933e-7, 0.1011704e-8, 0.100701e-9, 0.10133e-10]


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The normalized rational numbers R(2*n)/R(0) are listed as 
A096952(n)/ A096952(n).

[1, 7/108, 11/1296, 463/326592, 4039/15116544, 35839/665127936, 

320503/28298170368, 575267/235092492288, 25854247/47958868426752, 

232557151/1929639176699904, 298927153/10968475320188928, 

18830313487/3027299188372144128, 6778577311/4738381338321616896, 

1525146340543/4605706660848611622912, 13726182847159/178087324219479649419264, 

123535108753519/6853291511342734094893056]

and their values are approximately 


[1., 0.6481481481e-1, 0.8487654321e-2, 0.1417670978e-2, 0.2671907018e-3,

0.5388286683e-4, 0.1132592658e-4, 0.2446981588e-5, 0.5390920980e-6, 

0.1205184647e-6, 0.2725330042e-7, 0.6220169304e-8, 0.1430568126e-8, 

0.3311427437e-9, 0.7707557462e-10, 0.1802566089e-10]

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