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 y-axis, and with the j-th 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 k-th arc from (1, sqrt(k^2 - 1)) to (0, k).
Let alpha(k) be the angle covered by the k-th 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 Euler-Mascheroni 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 k-th arc approaches 1, and (sum of the lengths of the first k arcs) - k also approaches a constant; see A323737.)
FORMULA
Lim_{k->infinity} (Sum_{j=1..k} arcsin(1/j)) - log(k).
EXAMPLE
1.18490520294864502273935712638320241267802251776496...
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jon E. Schoenfield, Feb 03 2019
STATUS
approved