A323736 Decimal expansion of the limit of (Sum_{j=1..k} arcsin(1/j)) - log(k) as k approaches infinity.

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

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.)

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: A335354 A000803 A198063 * A093208 A155064 A338574

Adjacent sequences: A323733 A323734 A323735 * A323737 A323738 A323739

Keyword

nonn,cons

Author

Jon E. Schoenfield, Feb 03 2019

Status

approved