

A323736


Decimal expansion of the limit of (Sum_{j=1..k} arcsin(1/j))  log(k) as k approaches infinity.


2



1, 1, 8, 4, 9, 0, 5, 2, 0, 2, 9, 4, 8, 6, 4, 5, 0, 2, 2, 7, 3, 9, 3, 5, 7, 1, 2, 6, 3, 8, 3, 2, 0, 2, 4, 1, 2, 6, 7, 8, 0, 2, 2, 5, 1, 7, 7, 6, 4, 9, 6, 9, 0, 1, 5, 1, 4, 1, 5, 2, 2, 4, 3, 5, 7, 5, 1, 0, 5, 3, 0, 0, 0, 7, 4, 3, 7, 4, 0, 3, 7, 4, 0, 3, 1, 0, 4
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OFFSET

1,3


COMMENTS

Draw a set of circular arcs, with each arc having its center at the origin and drawn counterclockwise so that it connects the line x=1 to the yaxis, and with the jth arc having radius j; i.e., draw
 the 1st arc from (1, 0) to (0, 1),
 the 2nd arc from (1, sqrt(3)) to (0, 2),
 the 3rd arc from (1, sqrt(8)) to (0, 3),
...
 the kth arc from (1, sqrt(k^2  1)) to (0, k).
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), which approaches 1/k as k increases.
Let s(k) = Sum_{j=1..k} alpha(j) = Sum_{j=1..k} arcsin(1/j); since arcsin(1/j) > 1/j for j >= 1, and Sum_{j=1..k} 1/j diverges as k increases, s(k) must likewise diverge as k increases.
Lim_{k>infinity} ((Sum_{j=1..k} 1/j)  log(k)) approaches the EulerMascheroni constant, 0.5772156649015... (cf. A001620); this sequence gives the decimal expansion of lim_{k>infinity} (s(k)  log(k)).
(As k increases, the length of the kth arc approaches 1, and (sum of the lengths of the first k arcs)  k also approaches a constant; see A323737.)


LINKS

Table of n, a(n) for n=1..87.


FORMULA

Lim_{k>infinity} (Sum_{j=1..k} arcsin(1/j))  log(k).


EXAMPLE

1.18490520294864502273935712638320241267802251776496...


CROSSREFS

Cf. A001620, A323737.
Sequence in context: A124012 A000803 A198063 * A093208 A155064 A011225
Adjacent sequences: A323732 A323733 A323734 * A323737 A323738 A323739


KEYWORD

nonn,cons


AUTHOR

Jon E. Schoenfield, Feb 03 2019


STATUS

approved



