%I
%S 6,8,5,3,8,5,3,0,2,7,5,1,2,3,3,7,5,5,5,1,8,7,4,4,6,0,5,4,3,0,1,3,0,7,
%T 7,0,3,4,3,5,9,3,7,1,0,1,3,8,0,4,3,2,0,5,3,0,8,5,1,5,0,2,1,5,5,3,8,6,
%U 0,1,6,3,7,1,3,1,4,3,3,9,3,6,0,2,6,6,8
%N Decimal expansion of the limit of (Sum_{j=1..k} j*arcsin(1/j))  k as k approaches infinity.
%C Draw a set of circular arcs, with each arc having its center at the origin and drawn so that it connects the line x=1 to the positive yaxis, and with the jth arc having radius j; i.e., draw
%C  the 1st arc from (1, 0) to (0, 1),
%C  the 2nd arc from (1, sqrt(3)) to (0, 2),
%C  the 3rd arc from (1, sqrt(8)) to (0, 3),
%C ...
%C  the kth arc from (1, sqrt(k^2  1)) to (0, k).
%C Let alpha(k) be the angle covered by the kth arc: the first arc is a quarter circle, so alpha(1) = Pi/2; alpha(2) = arcsin(1/2) = 0.523598...; alpha(3) = arcsin(1/3) = 0.339836...; and, in general, alpha(k) = arcsin(1/k). The length of the kth arc is k*alpha(k) = k*arcsin(1/k), which approaches 1 as k increases.
%C The total length of the first k arcs is then t(k) = Sum_{j=1..k} j*alpha(j) = Sum_{j=1..k} j*arcsin(1/j), which equals k + C  (1/6)/k + (1/12)/k^2  (19/360)/k^3 + (3/80)/k^4  (143/5040)/k^5 + (5/224)/k^6  (731/40320)/k^7 + (35/2304)/k^8  (4949/380160)/k^9 + (63/5632)/k^10  ... where C = 0.685385... is the constant whose decimal expansion consists of the terms of this sequence.
%F Lim_{k>infinity} (Sum_{j=1..k} j*arcsin(1/j))  k.
%e 0.68538530275123375551874460543013077034359371013804...
%Y Cf. A323736.
%K nonn,cons
%O 0,1
%A _Jon E. Schoenfield_, Feb 06 2019
