Sum of the reciprocals of even-index Catalan numbers (A048990).
1, 5, 7, 9, 7, 6, 8, 3, 7, 9, 5, 5, 4, 0, 2, 0, 7, 7, 5, 2, 4, 2, 9, 9, 7, 8, 5, 9, 1, 2, 3, 4, 4, 4, 8, 6, 0, 6, 2, 7, 8, 9, 5, 5, 3, 5, 7, 6, 6, 4, 9, 5, 0, 5, 5, 2, 0, 7, 1, 8, 1, 8, 5, 4, 0, 1, 6, 9, 2, 3, 7, 9, 2, 9, 8, 4, 0, 7, 3, 6, 3, 6, 7, 5, 8, 6, 0, 3, 4, 4, 4, 9, 6, 4, 2, 3, 6, 1, 3, 7, 1, 1, 4, 9, 7
1
Koshy and Gao indicate how to derive the formula for the sum.
T. D. Noe, Plot of 1001 terms
T. D. Noe, Table of 1001 terms
Thomas Koshy and Zhenguang Gao, Convergence of a Catalan Series, Vol. 43, No. 2, March 2012, The College Mathematics Journal
Wikipedia, Catalan number
The number is 1.579768379554020775242997859123444860627895535766495….
(Mma) RealDigits[32/25 + 2*Sqrt[3]*Pi/27 - 12*Sqrt[5]*Log[GoldenRatio]/125, 10, 101][[1]]
nonn,cons
T. D. Noe, Aug 26 2014